Problems in Algebraic Number Theory

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Springer Science & Business Media, 2005 - Mathematics - 352 pages
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Asking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking "well-posed" questions. The approach taken by the authors in Problems in Algebraic Number Theory is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically arranged to reveal ideas and concepts in the evolution of the subject. While some problems are easy and straightforward, others are more difficult. For this new edition the authors added a chapter and revised several sections. The text is suitable for a first course in algebraic number theory with minimal supervision by the instructor. The exposition facilitates independent study, and students having taken a basic course in calculus, linear algebra, and abstract algebra will find these problems interesting and challenging. For the same reasons, it is ideal for non-specialists in acquiring a quick introduction to the subject.
  

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Contents

Elementary Number Theory
1
12 Applications of Unique Factorization
6
13 The ABC Conjecture
7
14 Supplementary Problems
8
Euclidean Rings
11
22 Gaussian Integers
15
24 Some Further Examples
19
25 Supplementary Problems
23
The Ideal Class Group
67
62 Finiteness of the Ideal Class Group
69
63 Diophantine Equations
71
64 Exponents of Ideal Class Groups
73
65 Supplementary Problems
74
Quadratic Reciprocity
79
72 Gauss Sums
82
73 The Law of Quadratic Reciprocity
84

Algebraic Numbers and Integers
25
32 Liouvilles Theorem and Generalizations
27
33 Algebraic Number Fields
30
34 Supplementary Problems
36
Integral Bases
39
42 Existence of an Integral Basis
41
43 Examples
44
44 Ideals in OK
47
45 Supplementary Problems
48
Dedekind Domains
51
52 Characterizing Dedekind Domains
53
53 Fractional Ideals and Unique Factorization
55
54 Dedekinds Theorem
61
55 Factorization in Ok
63
56 Supplementary Problems
64
75 Primes in Special Progressions
89
The Structure of Units
3
83 Supplementary Problems
19
Higher Reciprocity Laws
21
92 Eisenstein Reciprocity
26
93 Supplementary Problems
29
Analytic Methods
31
102 Zeta Functions of Quadratic Fields
34
103 Dirichlets LFunctions
37
104 Primes in Arithmetic Progressions
38
Density Theorems
43
112 Distribution of Prime Ideals
50
113 The Chebotarev density theorem
54
114 Supplementary Problems
57
Copyright

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Page v - What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."4 This is an expressive way of saying that you have added to the accumulation of your mathematical intuition.
Page v - Precepts, I have thought fit to adjoin the Solutions of the following Problems.

About the author (2005)

Ram Murty is a Professor and Queen's Reseach Chair at Queen's University.

Esmonde, McGill University, Montreal.

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