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Philosophy: What makes someone a person? -…

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Sample Essay Responses and Rater Commentary for **what makes**, the Argument Task.
The sample essays that follow were written in response to the prompt that appears below . The rater commentary that follows each sample essay explains how the response meets the criteria for **power**, that score. For a more complete understanding of the criteria for each score point, see the Analyze an Argument Scoring Guide.
In surveys Mason City residents rank water sports (swimming, boating and fishing) among their favorite recreational activities. The Mason River flowing through the city is rarely used for these pursuits, however, and the city park department devotes little of its budget to **makes a person** maintaining riverside recreational facilities. *On Senses And Place: Half An Hour*? For years there have been complaints from residents about the quality of the *what makes* river's water and the river's smell. In response, the state has recently announced plans to clean up Mason River. Use of the river for water sports is therefore sure to increase.

The city government should for that reason devote more money in this year's budget to riverside recreational facilities.
Write a response in heineken, which you examine the stated and/or unstated assumptions of the *a person* argument. *Essay On Senses Half At The*? Be sure to explain how the *what a person a person* argument depends on the assumptions and what the implications are if the assumptions prove unwarranted.
Note: All responses are reproduced exactly as written, including errors, misspellings, etc., if any.
While it may be true that the Mason City government ought to devote more money to riverside recreational facilities, this author's argument does not make a cogent case for increased resources based on river use. *The Tensions*? It is easy to **a person** understand why city residents would want a cleaner river, but this argument is rife with holes and assumptions, and thus, not strong enough to lead to increased funding.
Citing surveys of city residents, the author reports city resident's love of water sports. It is not clear, however, the *Ethical and Natural Sciences Essay* scope and validity of that survey. For example, the survey could have asked residents if they prefer using the river for water sports or would like to see a hydroelectric dam built, which may have swayed residents toward river sports. The sample may not have been representative of city residents, asking only those residents who live upon the river.

The survey may have been 10 pages long, with 2 questions dedicated to river sports. *What A Person*? We just do not know. Unless the *power distance* survey is fully representative, valid, and reliable, it can not be used to effectively back the author's argument.
Additionally, the author implies that residents do not use the river for swimming, boating, and fishing, despite their professed interest, because the water is polluted and smelly. *A Person A Person*? While a polluted, smelly river would likely cut down on river sports, a concrete connection between the resident's lack of river use and the river's current state is not effectively made. Though there have been complaints, we do not know if there have been numerous complaints from a wide range of people, or perhaps from one or two individuals who made numerous complaints. *Enhancing*? To strengthen his/her argument, the *a person* author would benefit from implementing a normed survey asking a wide range of residents why they do not currently use the river.
Building upon the implication that residents do not use the river due to **Essay Half an Hour at the Rooftop** the quality of the river's water and the smell, the author suggests that a river clean up will result in increased river usage. *What A Person A Person*? If the river's water quality and smell result from problems which can be cleaned, this may be true. For example, if the decreased water quality and aroma is caused by pollution by factories along the river, this conceivably could be remedied.

But if the quality and **heineken**, aroma results from the natural mineral deposits in the water or surrounding rock, this may not be true. There are some bodies of water which emit a strong smell of what sulphur due to the geography of the area. This is not something likely to be afffected by a clean-up. Consequently, a river clean up may have no impact upon river usage. Regardless of whether the river's quality is able to be improved or not, the author does not effectively show a connection between water quality and river usage.
A clean, beautiful, safe river often adds to a city's property values, leads to increased tourism and revenue from those who come to take advantage of the river, and a better overall quality of life for residents. *And Place: Half An Hour At The Rooftop*? For these reasons, city government may decide to invest in improving riverside recreational facilities. *Makes A Person*? However, this author's argument is not likely significantly persuade the city goverment to **Performance Drugs' on Athletes Essay** allocate increased funding.

Rater Commentary for **a person**, Essay Response Score 6.
This insightful response identifies important assumptions and **Essay Half an Hour**, thoroughly examines their implications. The essay shows that the proposal to **what makes a person** spend more on riverside recreational facilities rests on three questionable assumptions, namely:
that the survey provides a reliable basis for budget planning that the river’s pollution and odor are the only reasons for its limited recreational use that efforts to clean the water and **when was the townshend act passed**, remove the odor will be successful.
By showing that each assumption is *makes a person*, highly suspect, this essay demonstrates the weakness of the entire argument. For example, paragraph 2 points out that the survey might not have used a representative sample, might have offered limited choices, and might have contained very few questions on *index*, water sports.
Paragraph 3 examines the *what a person* tenuous connection between complaints and limited use of the river for **when townshend**, recreation. Complaints about water quality and odor may be coming from *makes a person a person* only a few people and, even if such complaints are numerous, other completely different factors may be much more significant in reducing river usage. Finally, paragraph 4 explains that certain geologic features may prevent effective river clean-up.

Details such as these provide compelling support.
In addition, careful organization ensures that each new point builds upon the previous ones. For example, note the clear transitions at the beginning of paragraphs 3 and 4, as well as the logical sequence of sentences within paragraphs (specifically paragraph 4).
Although this essay does contain minor errors, it still conveys ideas fluently. Note the effective word choices (e.g., rife with . *Distance*? . . assumptions and may have swayed residents). In addition, sentences are not merely varied; they also display skillful embedding of subordinate elements.
Since this response offers cogent examination of the argument and conveys meaning skillfully, it earns a score of 6.
The author of this proposal to increase the budget for Mason City riverside recreational facilities offers an interesting argument but to move forward on the proposal would definitely require more information and thought. *What A Person*? While the correlations stated are logical and **on Senses and Place: Half an Hour Rooftop**, probable, there may be hidden factors that prevent the City from diverting resources to this project.

For example, consider the *what makes a person* survey rankings among Mason City residents. The thought is that such high regard for water sports will translate into usage. But, survey responses can hardly be used as indicators of actual behavior. Many surveys conducted after the winter holidays reveal people who list exercise and **power distance**, weight loss as a top priority. Yet every profession does not equal a new gym membership. Even the wording of the survey results remain ambiguous and **what makes a person**, vague. While water sports may be among the residents' favorite activities, this allows for **heineken**, many other favorites.

What remains unknown is the priorities of the general public. Do they favor these water sports above a softball field or soccer field? Are they willing to sacrifice the municipal golf course for better riverside facilities? Indeed the survey hardly provides enough information to discern future use of improved facilities.
Closely linked to the surveys is the bold assumption that a cleaner river will result in what a person, increased usage. While it is not illogical to expect some increase, at **Sciences** what level will people begin to use the river? The answer to this question requires a survey to **what makes a person** find out the reasons our residents use or do not use the river. Is river water quality the primary limiting factor to usage or the lack of docks and piers?

Are people more interested in water sports than the recreational activities that they are already engaged in? These questions will help the city government forecast how much river usage will increase and to assign a proportional increase to **and Place: an Hour** the budget.
Likewise, the author is optimistic regarding the state promise to clean the *what makes a person a person* river. We need to hear the source of the *Changing 1920s* voices and **what**, consider any ulterior motives. Is this a campaign year and the plans a campaign promise from the state representative? What is the timeline for **when act passed**, the clean-up effort?

Will the state fully fund this project? We can imagine the misuse of funds in what, renovating the riverside facilities only to watch the new buildings fall into dilapidation while the *Half an Hour at the* state drags the river clean-up.
Last, the author does not consider where these additional funds will be diverted from. The current budget situation must be assessed to determine if this increase can be afforded. In a sense, the City may not be willing to draw money away from other key projects from road improvements to schools and education. The author naively assumes that the money can simply appear without forethought on where it will come from.
Examining all the various angles and **what**, factors involved with improving riverside recreational facilities, the argument does not justify increasing the budget. While the proposal does highlight a possibility, more information is *Essay and Place: Half an Hour*, required to **makes a person** warrant any action.
Rater Commentary for Essay Response Score 5.

Each paragraph in Essay and Place: an Hour, the body of this perceptive essay identifies and **what makes a person**, examines an Essay an Hour unstated assumption that is crucial to **what makes a person a person** the argument. The major assumptions discussed are:
that a survey can accurately predict behavior that cleaning the river will, in itself, increase recreational usage that state plans to clean the river will actually be realized that Mason City can afford to spend more on riverside recreational facilities.
Support within each paragraph is both thoughtful and **heineken international**, thorough. *A Person*? For example, paragraph 2 points out vagueness in the wording of the survey: Even if water sports rank among the favorite recreational activities of Mason City residents, other sports may still be much more popular. Thus, if the first assumption proves unwarranted, the argument to fund riverside facilities rather than soccer fields or golf courses becomes much weaker. Paragraph 4 considers several reasons why river clean-up plans may not be successful (the plans may be nothing more than campaign promises or funding may not be adequate). Thus, the weakness of the third assumption undermines the argument that river recreation will increase and riverside improvements will be needed at all.
Instead of dismissing each assumption in when was the townshend, isolation, this response places them in a logical order and considers their connections. *A Person A Person*? Note the appropriate transitions between and within paragraphs, clarifying the links among the assumptions (e.g., Closely linked to the surveys or The answer to this question requires. ).

Along with strong development, this response also displays facility with language. Minor errors in punctuation are present, but word choices are apt and sentences suitably varied in pattern and length. The response uses a number of rhetorical questions, but the implied answers are always clear enough to **on Athletes** support the points being made.
Thus, the response satisfies all requirements for a score of 5, but its development is not thorough or compelling enough for a 6.
The problem with the *makes a person* arguement is the assumption that if the Mason River were cleaned up, that people would use it for water sports and **in Art and Natural Sciences**, recreation. This is not necessarily true, as people may rank water sports among their favorite recreational activities, but that does not mean that those same people have the *what a person a person* financial ability, time or equipment to **power** pursue those interests.
However, even if the *a person* writer of the arguement is correct in assuming that the Mason River will be used more by the city's residents, the arguement does not say why the recreational facilities need more money. If recreational facilities already exist along the Mason River, why should the city allot more money to fund them? If the recreational facilities already in existence will be used more in the coming years, then they will be making more money for themselves, eliminating the need for **heineken international**, the city government to devote more money to **what** them.
According to the arguement, the *The Tensions 1920s Essay* reason people are not using the Mason River for water sports is because of the smell and the quality of water, not because the recreational facilities are unacceptable.

If the city government alloted more money to the recreational facilities, then the budget is being cut from some other important city project. *A Person A Person*? Also, if the assumptions proved unwarranted, and more people did not use the *heineken international* river for recreation, then much money has been wasted, not only the *makes a person* money for the recreational facilities, but also the money that was used to clean up the river to attract more people in power distance index, the first place.
Rater Commentary for Essay Response Score 4.
This competent response identifies two unstated assumptions:
that cleaning up the Mason River will lead to increased recreational use that existing facilities along the river need more funding.
Paragraph 1 offers reasons why the first assumption is questionable (e.g., residents may not have the necessary time or money for water sports). Similarly, paragraphs 2 and 3 explain that riverside recreational facilities may already be adequate and may, in fact, produce additional income if usage increases. *What A Person A Person*? Thus, the response is *international*, adequately developed and satisfactorily organized to show how the argument depends on questionable assumptions.
However, this essay does not rise to a score of 5 because it fails to consider several other unstated assumptions (e.g., that the survey is reliable or that the efforts to clean the river will be successful). *What Makes A Person A Person*? Furthermore, the final paragraph makes some extraneous, unsupported assertions of Drugs' on Athletes its own.

Mason City may actually have a budget surplus so that cuts to other projects will not be necessary, and cleaning the river may provide other real benefits even if it is *what a person a person*, not used more for water sports.
This response is generally free of errors in grammar and usage and displays sufficient control of language to support a score of 4.
Surveys are created to speak for the people; however, surveys do not always speak for the whole community. *Power*? A survey completed by Mason City residents concluded that the residents enjoy water sports as a form of recreation. If that is so evident, why has the river not been used? The blame can not be soley be placed on the city park department. The city park department can only do as much as they observe. The real issue is not the residents use of the river, but their desire for a more pleasant smell and a more pleasant sight. If the city government cleans the river, it might take years for the smell to go away. If the budget is changed to **what makes a person** accomodate the *and Place: at the* clean up of the Mason River, other problems will arise.

The residents will then begin to complain about other issues in a person, their city that will be ignored because of the great emphasis being placed on Mason River. If more money is *Essay on Senses Half an Hour Rooftop*, taken out of the budget to clean the river an makes a person a person assumption can be made. This assumption is *was the townshend*, that the budget for **a person**, another part of cit maintenance or building will be tapped into to. In addition, to the budget being used to **heineken** clean up Mason River, it will also be allocated in increasing riverside recreational facilites. The government is *what*, trying to appease its residents, and one can warrant that the *index* role of the *what a person* government is to please the people. There are many assumptions being made; however, the government can not make the assumption that people want the river to be cleaned so that they can use it for recreational water activities. The government has to **international** realize the long term effects that their decision will have on the monetary value of their budget.
Rater Commentary for Essay Response Score 3.
Even though much of this essay is tangential, it offers some relevant examination of the argument’s assumptions.

The early sentences mention a questionable assumption (that the survey results are reliable) but do not explain how the survey might have been flawed. Then the response drifts to irrelevant matters a defense of the city park department, a prediction of budget problems and the problem of pleasing city residents.
Some statements even introduce unwarranted assumptions that are not part of the original argument (e.g., The residents will then begin to complain about other issues and This assumption is that the *makes a person* budget for **on Senses and Place: Half at the**, another part of city maintenance or building will be tapped into). *A Person*? Near the end, the response does correctly note that city government should not assume that residents want to use the river for recreation. Hence, the proposal to increase funding for riverside recreational facilities may not be justified.
In summary, the language in this response is reasonably clear, but its examination of unstated assumptions remains limited and therefore earns a score of 3.
This statement looks like logical, but there are some wrong sentences in it which is not logical.
First, this statement mentions raking water sports as their favorite recreational activities at the first sentence.

However, it seems to have a ralation between the first sentence and the setence which mentions that increase the quality of the river's water and **power**, the river's smell. This is a wrong cause and result to solve the problem.
Second, as a reponse to the complaints from residents, the state plan to **a person a person** clean up the river. As a result, the state expects that water sports will increase. *Townshend*? When you look at two sentences, the result is *what*, not appropriate for **international**, the cause.
Third, the last statement is the *what makes* conclusion. However, even though residents rank water sports, the city government might devote the budget to another issue. This statement is also a wrong cause and **heineken**, result.
In summary, the statement is not logical because there are some errors in a person, it. *On Athletes Essay*? The supporting setences are not strong enough to support this issue.

Rater Commentary for Essay Response Score 2.
Although this essay appears to be carefully organized, it does not follow the directions for the assigned task. In his/her vague references to causal fallacies, the writer attempts logical analysis but never refers to any unstated assumptions. Furthermore, several errors in grammar and sentence structure interfere with meaning (e.g., This statement looks like logical, but there are some wrong sentences in it which is not logical).
Because this response does not follow the directions for the assigned task and **makes**, contains errors in Essay on Senses and Place: Half at the Rooftop, sentence structure and logical development, it earns a score of 2.
The statement assumes that everyone in Mason City enjoys some sort of recreational activity, which may not be necessarily true. The statement also assumes that if the state cleans up the river, the *what makes* use of the *Essay Half* river for water sports will definitely increase.

Rater Commentary for Essay Response Score 1.
The brevity of this two-sentence response makes it fundamentally deficient. Sentence 1 states an assumption that is actually not present in the argument, and **what makes a person**, sentence 2 correctly states an assumption but provides no discussion of its implications. Although the response may begin to address the assigned task, it offers no development. As such, it is clearly extremely brief . providing little evidence of an organized response and earns a score of 1.
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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. **What**. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in *in Art and Natural Essay* the number field, the ideals and units in the ring of integers, the extent to *what a person*, which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008).
Corrected; revisions and Essay Half an Hour, additions; 163 pages. v3.01 (September 28, 2008).

Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). **What Makes A Person**. Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne.
Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **Townshend**. . . . . 5 Prerequisites . . . . . . . . . . **What Makes A Person**. . . . . . . . . **Performance Enhancing Essay**. . . . . . . . **What**. . . . . **On Senses And Place: At The Rooftop**. . . . . . . . . 5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . **What Makes A Person A Person**. . . . . . . . 5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **The Tensions Essay**. . . . 1 Exercises . . . . . . . . . . . . . . . . . . . **Makes A Person**. . . . . . . . . . **On Senses And Place: An Hour At The Rooftop**. . . . . . **Makes A Person**. . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . **Performance Essay**. . . . . . . . . . . . . . . . . . . . . **What Makes A Person**. . . . 7 Ideals in products of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . . . . **Power**. . . . **What Makes A Person**. . . . . . . . . . . **Heineken**. . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . . . . . . . . **What Makes A Person A Person**. . . . . . . . . . . 10 Local rings . . . . . **Enhancing Essay**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Rings of fractions . . . . . . . . . . . . . **What Makes A Person A Person**. . . . . . . **Of The 1920s Essay**. . . . . **Makes A Person**. . . . . . . . . . . . **Was The Act Passed**. 11 The Chinese remainder theorem . . . . . . . . . . . . . **Makes**. . . . . . **The Tensions Essay**. . . . . **Makes**. . . . . **Power Distance**. 12 Review of tensor products . . . . . . . . . . . . . . . . . . . **What A Person**. . . . . . . . . . . . 14 Exercise . **Enhancing Drugs' Affects On Athletes**. . . . . . **What**. . . . . . . . . . . . . . . . . . **Of The Essay**. . **What Makes**. . . . . . . . . . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . . . . . . . . **Enhancing Drugs' Affects**. . . 19 Dedekind’s proof that the integral elements form a ring . . . . . . . . . . . . . **Makes**. . 20 Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **When Townshend**. . 22 Review of bases of A-modules . . . . **Makes A Person**. . . . . . . . . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . . . . . . . . . . . . . . **Changing**. . . . **What Makes A Person A Person**. . . **Heineken**. . **What Makes A Person**. . . . . . . 25 Review of bilinear forms . . . **At The Rooftop**. . . . . . . . . . . . . . . . . **A Person**. . . . . . . . . . . . 26 Discriminants . . . . . . . . . . **Index**. . **A Person**. . . . . . . . . . . . . . . . . . . . . . . **Distance**. . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . **A Person**. . . . . 29 Finding the ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **Ethical In Art**. 31 Algorithms for finding the **makes** ring of integers . **Of The Changing 1920s Essay**. . . . . . . . . . . . . . . **What Makes A Person**. . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . **Heineken International**. . . . . . . . . . . 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . **Makes A Person A Person**. . . . **Of The**. . . . . . . . . . . . . **Makes A Person A Person**. . . . . . . . . . . **Index**. . . . 40 Dedekind domains . . **What Makes A Person**. . . . . . . . . **Ethical Judgements In Art And Natural Sciences**. . . . . . . . . . . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . . . . . . . **What Makes A Person**. . . . . . **Heineken International**. . **What Makes A Person**. . . **On Senses And Place: Half Rooftop**. . **Makes A Person**. . . . . . **Essay On Senses Half**. . . . . 43 The ideal class group . **What Makes A Person**. . . . . . . **Of The**. . . . . . . . . . **What A Person**. . . . . . . . . . . . . . . . . 46 Discrete valuations . . . . . . . . . **Ethical Judgements In Art Sciences Essay**. . . . . . . . . . . . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . **Makes A Person A Person**. . . . . . . **Power Index**. . . **What Makes A Person A Person**. . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . . . . . . . . . . . . . . . . . 52.

Factorization in extensions . . . . . . . . . . . . . . . . . . . . **The Tensions Changing 1920s**. . . . . . . . . . 52 The primes that ramify . . . . . . **A Person**. . **On Senses And Place: Half An Hour At The**. . . . . **What A Person A Person**. . . . . . . . . . . . . . **Enhancing On Athletes**. . . . . . . . **Makes A Person**. 54 Finding factorizations . . . . . . . **Essay On Senses Half An Hour At The**. . . . . . . . . . . . . . **What Makes**. . . . . . . **Heineken International**. . . . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . . . **What A Person**. . . . **Distance Index**. . . . . . **What A Person A Person**. . . 57 Eisenstein extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Exercises . . . . . . . . . . . **Ethical Judgements Essay**. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . **What**. . . . . . . . . . . . . 63 Statement of the main theorem and its consequences . . . . . **The Tensions Of The Changing**. . . . . . . . **What Makes A Person A Person**. . . **On Senses And Place: An Hour At The Rooftop**. . . 65 Lattices . **What A Person**. . . . . . . . . . . . . **Essay And Place: Half**. . . . . . . . . . . . . . . . . . . . **Makes**. . . . . . . . **Essay And Place: Half An Hour**. 68 Some calculus . . . . . . . . . . . . **What Makes**. . . . . . . . . . . . . . . . . . . . . . . . . **Power Distance**. 73 Finiteness of the class number . . . . . . . **What Makes A Person**. . . . . . . . . . . . . . . . . . . . . 75 Binary quadratic forms . . . . . . . . **Was The Townshend Act Passed**. . **Makes A Person A Person**. . . . . . . . . . **Heineken International**. . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . **A Person**. . . . . . . . . . . . . **Essay On Senses And Place: Half At The Rooftop**. . . . **What A Person A Person**. . **Index**. . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . **What Makes A Person**. . . . . . . . . . . . . . . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . . . . . **Ethical Judgements And Natural Sciences**. . . . **Makes**. . . . . . . . . . **Performance Enhancing Drugs' Essay**. . 82 Computation of the rank . . . . . . . **What Makes**. . . . . **Of The Changing 1920s Essay**. . . . . . . . . . . . **What A Person**. . . . . . . . . 83 S -units . . . . . . . . . . . **The Tensions Of The Changing**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Example: CM fields . . . . **What**. . . . . . . . . . . . . . **Heineken International**. . . . . . . . . . . . . . . . . 86 Example: real quadratic fields . . . . . . **Makes A Person**. . . **Distance**. . . . . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . **Makes A Person A Person**. . . . . . . . . . . . . 87 Finding .K/ . . . . . . . . . . . . . **Distance Index**. . . . . . . . . . . . . . . . . . . . . . . **What Makes A Person**. . 89 Finding a system of fundamental units . . . . **Power Distance**. . . **What A Person A Person**. . . . . **Was The Townshend**. . . . **What A Person**. . . . . . . . . **Power Distance**. . . 89 Regulators . . . . . . . **Makes A Person**. . . . . . . . . . . . . . **Power Distance**. . . . . . . . **Makes**. . . . . **Townshend**. . . . . . . . 89 Exercises . . . . **What Makes A Person**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **International**. . . . 90.
6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . **A Person**. . . . . . . **Ethical And Natural Essay**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **A Person**. . 97 The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . . . **When Was The**. 98 Exercises . . . . . **A Person**. . . **When Was The Townshend**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **A Person A Person**. . 100. 7 Valuations; Local Fields 101 Valuations . . . . **The Tensions Of The Changing 1920s**. . . . . . . . **What A Person**. . . . . . . . . . . . . . **Essay And Place: An Hour At The Rooftop**. . . . . . . . . . . . . . . 101 Nonarchimedean valuations . . . . . . . **What A Person A Person**. . . . . **Act Passed**. . . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . **What A Person**. . . . . **Was The Townshend Act Passed**. . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Complete list of valuations for what a person the rational numbers . . . . . . . . . . . . . . . . 105 The primes of a number field . . . . . . **Distance Index**. . . . . . . . **What A Person**. . . **International**. . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . . . . . . . . **What A Person A Person**. . . . . . . **Performance Drugs' On Athletes Essay**. . . . . . 109 Completions . . . . . . . . . . . . . . **What**. . . . . . . . . . . . . **Power Distance Index**. . . . . . . . . . **Makes A Person A Person**. . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . **Ethical Essay**. . . . 111 Newton’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . **Makes A Person A Person**. . . . . . . . . . . . . . **Distance**. . . . . . . . 118.

Newton’s polygon . . . . . . **A Person**. . . . **Essay An Hour**. . . . . . . . . . . . . . . . . . . . . . . . . . 120 Locally compact fields . . . **Makes A Person A Person**. . . . . . . . . . . . . . . . . **Enhancing Essay**. . . . . . . . . . . . . 122 Unramified extensions of a local field . . . **What Makes**. . . . **Heineken**. . . . . . . . . **Makes A Person**. . . . . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . . **Sciences**. . . . . . . . . 125 Ramification groups . . . . . . . . . . . . . . . . . . . . . **What**. . . . . **Of The 1920s**. . . . . . **A Person**. . . . 126 Krasner’s lemma and applications . . . **Of The Changing**. . . . **Makes A Person**. . . . . . . . . . . . . . . . . **Heineken International**. . . . 127 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . **Makes A Person A Person**. . . **Power Index**. . . **What**. . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . **Judgements In Art Sciences Essay**. . . . . . . **Makes A Person**. . . . . . . . . . . . 131 The product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . **Of The Essay**. . . . . . . . . . . . . . . . . . . 135 The Frobenius element . **A Person**. . . . . . . . . . . . . . . . . . . . . . . . . . . **Heineken International**. . . . . 137 Examples . . . . . . . . . . . . . . . . **Makes A Person A Person**. . . . **Distance Index**. . . . . . . . . . . . . **Makes A Person**. . **Distance Index**. . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . **What A Person A Person**. . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . **International**. . . . . . . **What Makes A Person A Person**. . . . **Power Distance**. . . . . 146 Exercises . . . **What A Person**. . . . . . . **Heineken**. . . **What A Person A Person**. . . . . . . . . . . . . . . **Was The Townshend Act Passed**. . . . . . . . . **What Makes**. . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that m divides n, i.e., n 2mZ.
Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. **Performance Affects On Athletes**. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and makes a person, Y are canonically isomorphic (or there is *The Tensions of the Changing 1920s*, a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. **Makes**. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q.

The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at Judgements in Art Sciences Essay the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011. GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and makes a person, comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an **when**, open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for what makes a person a person mD 4. He also posed the problem of finding integer solutions to the equation,
X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?.

The English mathemati- cians found an algorithm for solving the **distance index** problem, but neglected to prove that the algorithm always works. EULER (1707–1783). **A Person**. He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes,
and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (1752–1833). He introduced the **power distance** “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law.
He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and what makes, 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810–1893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes.

HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of **Drugs' Affects on Athletes Essay** a polynomial of degree 5 can be expressed in terms of elliptic functions. EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).

RIEMANN (1826–1866). Studied the Riemann zeta function, and a person, made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion.
DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes. Defined the zeta function of a number field. WEBER (1842–1913). **Ethical And Natural Essay**. Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941).

He gave the first definition of the field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943).
He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of **what makes a person** his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of **Judgements Essay** abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). **What A Person**. Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields. HECKE (1887–1947).

Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions.
ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. **Drugs' Affects On Athletes Essay**. HASSE (1898–1979).

He gave the first proof of local class field theory, proved the **makes** Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over **Performance on Athletes Essay**, number fields. WEIL (1906–1998).
Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925– ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields.
LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to *what makes a person*, lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba).
I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to *when was the townshend*, order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to *a person*, what extent unique factorization holds, or fails to *Judgements in Art Essay*, hold, in number fields. Three problems present themselves.

First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. **What Makes A Person**. Finally, since factorization is only considered up to units, in *index* order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation.
?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the **what** algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is *Essay*, a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. **Makes**. 2a 2 Z; a2?b2d 2 Z:
From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?. Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P.

as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain?

No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is *and Place: Half an Hour at the Rooftop*, true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have. 6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?.
p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for makes ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and.

1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. **When Townshend Act Passed**. a Noetherian ring), factorization is unique if all irreducible elements are prime.
What can we recover? Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the **makes a person** set of “prime numbers” so that, for example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”.

How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides.
Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in *distance* addition division by **a person**, a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”. Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for distance index all b 2OK I.
it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/
then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime.

For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so. Z? p ?5?=.3;1?.
which is an integral domain. How far is this from *a person* what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of **heineken** “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index.

The index is called the class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units. For example, .1C p 2/ is a unit of infinite. **What A Person**. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and heineken international, so. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that.
OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3.
The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that.

Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction. Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?.
Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. **Makes**. Some comments on the literature.
COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY.

Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. The notes in *Essay Half at the* Narkiewicz 1990 document the origins of most significant results in algebraic number theory. Lemmermeyer 2009, which explains the origins of **makes a person a person** “ideal numbers”, and other writings by **act passed**, the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is as described.
0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative.

Many results that were first proved for rings of integers in number fields are true for more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an **what makes**, element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the **on Athletes** identity element to the identity element.
A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. **Makes A Person A Person**. We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X. ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . **Ethical And Natural Sciences Essay**. It consists of finite sums. P ciai , ci 2 A, and it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. **Makes A Person A Person**. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in *heineken international* A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . **A Person A Person**. ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA!

A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is *Essay*, prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the **what makes** quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is *Changing*, a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is *makes a person*, maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field.
Every proper ideal a of A is contained in a maximal ideal — if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorn’s lemma.

In particular, every nonunit in *Essay* A is contained in a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in products of rings. PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the form.

pB (p a prime ideal of A), Ap (p a prime ideal of B). PROOF. Let c be an **what makes a person a person**, ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. **The Tensions Of The 1920s**. REMARK 1.2 The lemma extends in *a person* an obvious way to a finite product of **Performance Affects** rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j:
A ring A is Noetherian if every ideal in A is finitely generated. **Makes**. PROPOSITION 1.3 The following conditions on *act passed* a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . **What Makes A Person A Person**. (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in *The Tensions of the 1920s* S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and what a person, so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc..
From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/.

If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of Hilbert states that k?X1; . **When Act Passed**. ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of **makes a person** ideals. never becomes constant. PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements.

PROOF. We shall need to use that, for Enhancing elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and what makes a person, only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and The Tensions 1920s, d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. **Makes A Person**. From the **Enhancing Drugs'** maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2.
REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an **a person a person**, algebraic integer ? can not be an irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated.

PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is *on Senses Half an Hour at the*, Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. **Makes A Person**. PROOF. Similar to the proof of Proposition 1.3.
2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant.
Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring.

Then every finitely generated A-module is Noetherian. PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by **Performance Enhancing Essay**, induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by **makes a person**, a single element, the statement follows from (1.7).
2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in *on Senses at the Rooftop* A. Let M be a finitely generated A-module, and define. **What Makes A Person**. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is *when was the townshend*, a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is *what makes*, a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write.
aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions.

Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. For example, Q is the field of fractions of Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. **Essay On Senses And Place: An Hour Rooftop**. A subset S of **makes a person a person** A is *Ethical Judgements*, said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define. S?1AD fa=b 2K j b 2 Sg:
It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is *makes a person*, a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of **Enhancing Affects** d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg:

2This notation conflicts with a later notation in which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an integral domainA and a multiplicative subset S ofA. **What Makes**. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of **on Senses and Place: an Hour at the** S?1A aec D a if a is *a person*, a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in S?1A.
For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is *index*, prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of **what a person** prime ideals in S?1A; the inverse map is p 7! pA. PROOF.
It is *was the act passed*, easy to *a person*, see that. p a prime ideal disjoint from *index* S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse.

2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in *what makes a person a person* general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of A not containing tg. fprime ideals of A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem.
Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in *heineken* pairs; then for makes a person any integers x1; . ;xn, the congruences. The Chinese remainder theorem. **Was The Townshend Act Passed**. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . **What A Person**. Q mi /, which equals. Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have.
Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for heineken any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the **what a person** form xC a with a 2.
Q ai . In other words, the. natural maps give an exact sequence.

PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in a1C. **Was The Townshend Act Passed**. Q i2 ai , and what a person, so. We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that.
y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements.
1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2.

We complete the **when was the act passed** proof by induction.
This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an **what makes a person a person**, A-module. There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/
f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into *act passed*, f 0 D ? ?f with ?WQ!

P A-linear. The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute). We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and makes, as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is *Performance Affects Essay*, a canonical isomorphism.
Review of tensor products.

It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is *what makes a person*, a vector space over **Enhancing on Athletes**, k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. **What**. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that. REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. **Ethical Sciences Essay**. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN.
PROOF. Recall that M 0?N 0 is *a person*, generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. **Heineken International**. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and a person, som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it.

2. One can also show that if M 0!M !M 00! 0.
is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module. Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. **On Senses Half At The**. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years.

1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is.
P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. **What Makes A Person A Person**. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the **when townshend** theorem. When we tensor the isomorphism.
with M , we get an isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.

If A! B is an **what a person**, A-algebra and Affects on Athletes Essay, M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS.

If f WA! B and gWA! C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. **Makes**. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs.
We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and of the Changing 1920s, let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for a person K=k, then the **Ethical Judgements in Art and Natural** image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. **What A Person A Person**. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. It was not until the 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on your website. If you don't receive any email, please check your Junk Mail box.

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Essay: WHAT IS CRIME? Crime prevention and what a person, crime reduction.
Crime is any action or offence that defies a state or country and is punishable by law. **Enhancing Drugs'**! Crime has many definitions. In fact the most common thing about these definitions is that crime is punishable.

Crime cuts across many disciplines such as sociology, psychology and criminology. Each of these disciplines try to explain why crime is committed and how people are compelled to commit crime, a good example is sociology. Sociology attributes crime due to poor socialization in society, while psychology attributes crime mainly due to biological and Pathological criminogenic behaviors. **What A Person A Person**! Many scholars have tried to define crime and each has given many reasons why crime is committed. Scholars such Cesare Lombroso attribute crime to biological anomalies while scholars like Edwin Sutherland claim that criminal behavior is learned.

Generally all these come, to the same conclusions that crime is an offence punishable by power, law. **Makes**! There are two main types of **was the townshend** crime, these include violent crimes and property crime. Violent crime constitutes when someone decides to **what a person**, harm, threaten and conspire against someone else while property crime constitute someone who damages, destroys or steals someone’s property. Both violent and heineken, property crimes are offences which involve force and damage to society. There are different types of **makes a person a person** punishing crime, the was the townshend act passed, most common typologies are retribution, restorative justice, general and specific deterrence, rehabilitation and just deserts. Crime punishment has been there since the beginning of **makes a person** time, theoldesttype of punishment was retribution. A good example of how retribution justice was used was during the Hammurabi period. In those days if crime was committed it constituted an eye for an eye. **Enhancing Drugs' Affects Essay**! If I killed someone my punishment would be death. No one was spared.

Justice was viewed differently. In the recent times retribution has been reviewed and has been lowered to just deserts. The punishment is still harsh but considers many factors at hand, such as the state of mindof the offender. Crime has been there for a long time and has been defined and been punished in different ways. What constitutes a crime has also been reviewed .what was viewed a crime in the previous times is not a crime now. A good example is what a person, freedom of worship. Many people were not allowed to worship any other gods and did it secrecy due to fear of prosecution and being labeled a heretic. **The Tensions Of The 1920s Essay**! In present times one is allowed to worship any god and believe in whoever they please. Generallycrime is a wide topic and a person, has been vigorously studied in different aspects butin this essay I am going to focus mainly on the major objectives of crime prevention, typologies of crime reduction, law enforcement and power distance index, crime, recidivism of crime and interventions on reduction of **makes a person a person** crime.
2.0 OBJECTIVES OF CRIME PREVENTIONAND CRIME REDUCTION.

Crime prevention includes reducing and deterring crime and when was the act passed, criminals from **makes a person a person**, committing crimes. Crime reduction is quite similar to **Judgements in Art and Natural**, crime prevention, for crime reduction to occur we need to **what**, prevent it at *heineken* first. Crime prevention strategies are usually implemented by makes a person, criminal justice agencies, individuals, businesses and non-governmental agencies in order to maintain order and in Art and Natural Sciences Essay, enforce the law. Crime prevention strategies not only deter crime but also reduce the what makes a person a person, risk of increasing victimization in the society.Crime prevention has many objectives but the most main objective is to reduce and deter crime. **Was The**! Many criminal justice agencies have developed strategies through public policy in order to prevent crime. Various models have been adopted by makes a person, countries in order to combat crime. Kenya for example has enforced the Nyumbakumi initiative (community policing) spear headed by Kaguthi in order to combat crime.

By this strategy neighbors are supposed to be readily aw e and watchful of what happens in the neighborhood in order to deter criminals from committing crimes. There are many approaches of crime prevention; the main objectives have been included in these strategies. **Changing 1920s Essay**! These strategies are situational crime prevention strategy, environmental crime prevention, social crime prevention, developmental crime prevention, policing strategies, and what, community crime prevention strategies.
The environmental prevention strategy was first introduced by C. Ray Jeffery a criminologist. Environmental crime prevention strategy main objective is to protect the environment which entails wildlife, Nature and the atmosphere. Environmental crime entails an illegal act that harms the environment. Many international bodies such as Interpol and the UN have recognized environmental crime due to the havoc it has causedthe environment, Types of environmental crime may include dumping hazardous waste in the ocean, illegal wild life trade of **on Senses an Hour at the Rooftop** endangered species, smuggling, emitting chemicals those ozone layer and illegal logging of trees.

There many crimes associated with environmental crime but I am going to focus on the two main which affect many countries which is illegal trade of wildlife and logging of **a person** tress.
Many counties have been trying to fight this crime. Many influential people have actually fought against environmental crime and have actually received Nobel prizes for it. The late Wangari Maathai who was an *international*, activist for the environment was highly against illegal logging of trees. In fact she proposed that for **makes a person a person**, every tree that was cut down, three should beplanted. Prevention strategies have been implemented in **The Tensions Changing Essay**, order to combat crime. In Ireland under the department of agriculture section 37 of the forestry act. **Makes A Person A Person**! It is illegal to uproot any tree over ten years old or cut down any tree of any age (agriculture, 2015). Illegal wildlife trade is also a major problem. Kenya has had this problem for years, being one of the countries that harbors endangered species such as the white rhino and elephants. It has faced a lot of problems in trying to combat this problem.

Many poachers are killing these animals and selling the tusks of **Drugs' Affects on Athletes Essay** these animals for high prices. Elephant poaching was made illegal in 1973, and hunting without a permit in 1977. Kenya has roughened sentencing through increasing fines.Poachers caught with illegal wildlife such as tusks face fines up to **what a person a person**, 10 million Kenya shillings and was the act passed, jail time of 5 years(Kahumbu. 2013).Though it is still rampant prevention strategies have been implemented.
Situational crime prevention strategy was a concept that gained wide recognition in the late 1940’s when Edwin in **makes a person**, Sutherland argued that crime was a result of environmental factors. Hebelieved that crime was learned. Situational crime prevention strategy is deeply rooted in theories such as routine activity theory, crime pattern theory and rational choice theory. **Heineken**! Situational crime prevention strategy focuses on mainly reducing crime by providing settings in which it is less conducive for criminals to **what a person**, attack.

Unlike routine, rational and crime prevention theories, situational prevention theory not only townshend act passed, focuses on the criminals but focuses mainly on the environment. **What**! A good example of how criminal justice agencies have applied this strategy is by Ethical Judgements in Art and Natural Sciences, ensuring that their heavy surveillance in **a person**, the cities in **Performance Drugs' Affects on Athletes Essay**, order to deter criminals from committing crimes. In Kenya the Government has installed cameras on the traffic lights in order to record criminal activity and find corrupt road traffic users (Okere, 2012). The Cameras not only deter people from committing crimes but also helps the police to .find culprits who may commit a crime and get away with it. A study done in Nairobi by Stephen Okere found out that 85.7% of all the what makes a person a person, Kenyans respondents of the Essay on Senses Half at the Rooftop, study had installed CCTV cameras and makes a person a person, found it effective in curbing crime. He also found that the traffic cameras also helped in curbing crime (Okere, 2012).The main objective of this crime prevention strategy isto protect people from criminals through providing or ensuring there are safety measures such as surveillance cameras.
Social crime prevention is a strategy that addresses the direct root causes of crime. **Essay On Senses And Place: Rooftop**! The main objective of social crime prevention is on the social elements that have lead people to commit this crimes, these elements may include breakdown in familyvalues and ignorance. Lack of cohesion and environmental conditions. Social crime prevention is not an easy task to achieve because it deals with peoples ideals bad believes.

The only way to create a society that is peaceful is to start from the beginning. This means ensuring that schooling from young age is given much importance. A good example of how governments have done this is by what a person a person, ensuring that the distance index, curriculum in nursery schools teaches children values of **what a person** what wrong and what is and Place: at the Rooftop, right. There are many ways of how social crime prevention can be achieved, through changing values at home through public education and encouraging the community to be the agent of social change in their own communities.
Developmental crime prevention focuses on how crime occurs; the mainobjective of this strategy is show how crime develops and causes victimization in **what a person**, society. **In Art And Natural Essay**! Developmental crime prevention strategy is used by many countries. Public education is one of the approaches that have been used. By using public education many people are taught and what makes a person, developed in to young abiding citizens rather than criminals.

Communities may also focus on helping teachers to be an integral part in **international**, developing self-control in young people. In the what, USA most stateshave developed programs which develop ex offender or drug addicts in to better people. They engage in social programs and help them achieve GEDS in order to get a better life. In general development crime prevention actually rehabilitates youth and helps develop others become better people rather than committing crime.
Policing strategies are also crucial in crime prevention. **Of The Essay**! The main objective of policing in crime prevention is to ensure that police officers actually do help citizens and actually, curb crime beforeit occurs. Policing should be proactive. When police actually improve on how they combat crime it helps reduce crime. **Makes A Person A Person**! Though police officers may be reluctant to change their ways, but with additional training they can change.

In order to reduce crime policing should be an important aspect. **Was The Townshend**! Community Crime prevention strategies are also important in curbing crime. The main objective of this strategy is to ensure that the community and police actually work together in order to prevent crime. By the community being involved in everything it helps reduce crime. **A Person**! Most countries have actually adopted this model. Kenya for example calls it nyumba kumi while other countries regard it as community policing. By the community and the police being involved it helps curb crime because the police are not working alone but are working hand in hand to ensure safety. Community crime prevention strategy can be very effective if the relationship between the citizen and the police is cordial.

If it is not, this approach can be very hard to achieve.
By societies using all these models of crime prevention, reduction of crime actually occurs. Crime reduction cannot occur if the government and criminal justice agencies are not doing anything about it. If you look at *international* countries that have high crime, the makes a person, criminal justice agency and Essay Half an Hour Rooftop, government are weak, and corruption is common. Such countries are run by cartels who engage in organized crime. Organized crime also tends to be present in **what a person**, countries that have strong criminal justice systems, but the difference between the two is that they are not strong as they are in **Affects**, failed states or weak countries. Guinea-Bissau for example which faces a lot corruption has made it easier for organized crime flourish. In April 2007 the authorizes of Guinea-Bissau managed to seize 635 kilograms of cocaine , unfortunately the drug traffickers managed to escape with 2.5 tons of **makes a person** drugs because the Performance Enhancing Affects on Athletes, police could not catch up with them (Mutume, 2007). The drug traffickers could have been captured but because of corruption and a poor criminal justice system the drug traffickers were able maneuver out with more than half.
Crime prevention and limitations.

Crime analysis is understood as the systematic study of **what makes a person a person** crime and of the Changing, disorder problems as well as other police-related issues (Santos). **What A Person A Person**! It is important to **index**, include sociodemographic, spatial, and mundane factors to assist in criminal apprehension, crime reduction, and what, crime prevention. It is used primarily as information so that personnel, from patrol officers to police chiefs, have an idea of when and where crime is Performance Drugs' on Athletes Essay, occurring and makes, how much it has overall occurred. While analysis has proven helpful in many cases, what it fails to do is directly inform proactive crime reduction strategies. This is because police officers are limited ion dealing with prevention. They are often assigned to **when act passed**, patrol areas where they are not fully familiar with.

They may not fully understand the social structure and norms that fuel the neighborhood and the actions of its residents.
While crime analysis was once focused primarily on tactical issues of **what a person** identifying offenders, discrimination and stereotyping led to social unrest and led to other tactics of **international** crime prevention. **What A Person**! With the stop and frisk campaign in new York, where the police had the heineken international, right to **what makes a person**, stop an individual and frisk them for any sort of weapons, drugs or paraphernalia, it became apparent hat innocent young blacks were not being targeted, but were having their rights infringed upon. This emphasizes the social and cultural disconnect between crime analysts, the sworn personnel, and the civilians they are attempting to protect. These became a blurred line between the officers’ role of protecting and when townshend, harassing innocent civilians. The question still remains how to effectively prevent and reduce crime.
Crime analysis and crime mapping are becoming more common, but they are primarily implemented in larger police agencies. Areas that have statistically needed more protection have been given more policing depending on the capacity of the police in **what makes a person**, the district. For example, it is argued tat there is a need for more policing in **heineken international**, urban areas because that is where crime is usually more prevalent, but that leaves other low population, yet crime ridden areas with less assistance. Despite this all, policing is occasionally being shifted to focus more on ‘hot spots,’ areas where crime is more prevalent.

The close monitoring has o an extent been able to deter crime, but that again depends on **what makes a person** the stance of the offender and what they have to lose from **The Tensions Changing 1920s**, their potential criminal transaction.
While in an ideal world all crime prevention efforts would work, that is what makes, not the Sciences Essay, case in the society that we live in today. Crime and its prevention vary depending on the environment of where the crime is happening. The demographics, the socioeconomic status of the people, and the relationships within the a person a person, community all factor into crime and its prevention. To address crime rates there must be various forms of **Ethical and Natural Sciences** prevention attempts. From the research conducted, it is evident that incarceration is limited in its effectiveness of **what a person** crime prevention and Performance Enhancing on Athletes, reduction. While there may be fewer criminals on the streets from incarceration, this does not directly affect rising crime rates. Given that about two thirds of criminals in the U.S. return to prison, incarceration only proves to be a temporary fix. I believe that incarceration would be more effective if there are efforts made in prison to better the lives of those incarcerated. Through efforts such as education, creating job skills and community buildings, those incarcerated are les likely to return to their former criminal past. This has the ability to create crime prevention and reduction in the long run.

I also believe that random patrol and reactive arrests used responses to a community’s demand are generally effective, policing in areas where crime is more prevalent makes it easier to identify problems within a community. It develops tailored responses in a timely manner so that crime can be controlled, reduced, and prevented.
I see various issues in maintaining prevention, the main one being sustainability. **A Person**! Prevention takes long-term planning with targeted spending and strong correspondence. It requires consistent community action and persistence with or without the presence of government funding. Without flexibility crime cannot be prevented or reduced. **Act Passed**! Like I have mentioned before, there are no two communities alike so there cannot be any single approach to sustainability. **What Makes A Person**! It is up to the individual communities and and Natural Essay, organizations to determine appropriate strategies and implement them. I agree with the makes a person, World Health Organization and the understanding that creating and implementing and monitoring a national action plan for violence prevention would be effective.

In order to do so, the issues of funding must be addressed. I believe that the federal and local government should invent in testing method of **heineken international** policing in order to raise awareness and reduce crime. To keep time rates low, there is a need to enhance the capacity of data collection on violence. That way, the issues that need to be addressed are apparent. When looking at issues and crimes within a community, it is what makes a person a person, important to **Ethical Judgements Sciences**, examine the causes. Consequences and makes a person a person, costs for prevention as well as reduction.
To keep crime prevention low, criminals as well as victims should be dealt with. By strengthening responses for **distance index**, victims, I believe that there will be a deterrence effect for criminals and less retaliation crimes that promote even more crime. I also believe that integrating crime prevention into social and educational policies has the ability to reduce crime by promoting social equality.
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