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E-raamat: Algebraic Number Theory for Beginners: Following a Path From Euclid to Noether

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Aug-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781009019224
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Algebraic Number Theory for Beginners: Following a Path From Euclid to NoetherSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Aug-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781009019224
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This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

This book, meant for undergraduate mathematics students and teachers, introduces algebraic number theory through problems from ordinary number theory that can be solved with the help of algebraic numbers, using a suitable generalization of unique prime factorization. The material is motivated by weaving historical information throughout.

Arvustused

'In Algebraic Number Theory for Beginners, John Stillwell once again displays his remarkable talent for using the history of mathematics to motivate and explore even the most abstract mathematical concepts at an accessible, undergraduate level. This book is another gem of the genre Stillwell has done so much to enhance.' Karen Hunger Parshall, University of Virginia 'Stillwell, more than any author I know, helps us understand mathematics from its roots. In this book, he leads us into algebraic number theory along a historical route from concrete to abstract. In doing so, Stillwell makes a strong pedagogical case for flipping a typical algebraic number theory course that students will understand number theory better if questions about numbers come before and throughout the abstract theory of rings and ideals. The treatments of mathematics and its history are crystal clear and meticulous. Stillwell's text is particularly well-suited for an advanced undergraduate or early graduate-level course in number theory. Experts also will find this text to be an incredible resource for its historical approach and well-motivated exercises. Stillwell has written another gem, this time for readers interested in number theory, abstract algebra, and their intertwined history.' Martin Weissman, University of California, Santa Cruz 'This book is sure to be welcomed by advanced students and their instructors A helpful index and an extensive list of references conclude the text Highly recommended.' J. Johnson, Choice 'It goes without saying that the exposition is as clear as possible. There are a few exercises in each section, historical comments, a decent bibliography and an index.' Franz Lemmermeyer, zbMATH Open

Muu info

A concise and well-motivated introduction to algebraic number theory, following the evolution of unique prime factorization through history.
Preface xi
Acknowledgments xiv
1 Euclidean Arithmetic
1(32)
1.1 Divisors and Primes
2(3)
1.2 The Form of the gcd
5(3)
1.3 The Prime Divisor Property
8(2)
1.4 Irrational Numbers
10(3)
1.5 The Equation x2 -- 2y2 = 1
13(2)
1.6 Rings
15(4)
1.7 Fields
19(3)
1.8 Factors of Polynomials
22(2)
1.9 Discussion
24(9)
2 Diophantine Arithmetic
33(26)
2.1 Rational versus Integer Solutions
34(2)
2.2 Fermat's Last Theorem for Fourth Powers
36(2)
2.3 Sums of Two Squares
38(3)
2.4 Gaussian Integers and Primes
41(2)
2.5 Unique Gaussian Prime Factorization
43(2)
2.6 Factorization of Sums of Two Squares
45(2)
2.7 Gaussian Primes
47(1)
2.8 Primes that Are Sums of Two Squares
48(2)
2.9 The Equation y3 = x2 + 2
50(3)
2.10 Discussion
53(6)
3 Quadratic Forms
59(19)
3.1 Primes of the Form x2 + ky2
60(1)
3.2 Quadratic Integers and Quadratic Forms
61(2)
3.3 Quadratic Forms and Equivalence
63(3)
3.4 Composition of Forms
66(2)
3.5 Finite Abelian Groups
68(3)
3.6 The Chinese Remainder Theorem
71(2)
3.7 Additive Notation for Abelian Groups
73(1)
3.8 Discussion
74(4)
4 Rings and Fields
78(26)
4.1 Integers and Fractions
79(3)
4.2 Domains and Fields of Fractions
82(1)
4.3 Polynomial Rings
83(3)
4.4 Algebraic Number Fields
86(3)
4.5 Field Extensions
89(4)
4.6 The Integers of an Algebraic Number Field
93(3)
4.7 An Equivalent Definition of Algebraic Integer
96(3)
4.8 Discussion
99(5)
5 Ideals
104(22)
5.1 "Ideal Numbers"
105(3)
5.2 Ideals
108(3)
5.3 Quotients and Homomorphisms
111(2)
5.4 Noetherian Rings
113(3)
5.5 Noether and the Ascending Chain Condition
116(3)
5.6 Countable Sets
119(2)
5.7 Discussion
121(5)
6 Vector Spaces
126(23)
6.1 Vector Space Basis and Dimension
127(3)
6.2 Finite-Dimensional Vector Spaces
130(4)
6.3 Linear Maps
134(2)
6.4 Algebraic Numbers as Matrices
136(3)
6.5 The Theorem of the Primitive Element
139(3)
6.6 Algebraic Number Fields and Embeddings in C
142(2)
6.7 Discussion
144(5)
7 Determinant Theory
149(22)
7.1 Axioms for the Determinant
150(3)
7.2 Existence of the Determinant Function
153(3)
7.3 Determinants and Linear Equations
156(3)
7.4 Basis Independence
159(2)
7.5 Trace and Norm of an Algebraic Number
161(3)
7.6 Discriminant
164(4)
7.7 Discussion
168(3)
8 Modules
171(18)
8.1 From Vector Spaces to Modules
172(2)
8.2 Algebraic Number Fields and Their Integers
174(2)
8.3 Integral Bases
176(3)
8.4 Bases and Free Modules
179(3)
8.5 Integers over a Ring
182(2)
8.6 Integral Closure
184(2)
8.7 Discussion
186(3)
9 Ideals and Prime Factorization
189(22)
9.1 To Divide Is to Contain
190(2)
9.2 Prime Ideals
192(2)
9.3 Products of Ideals
194(2)
9.4 Prime Ideals in Algebraic Number Rings
196(1)
9.5 Fractional Ideals
197(2)
9.6 Prime Ideal Factorization
199(2)
9.7 Invertibility and the Dedekind Property
201(3)
9.8 Discussion
204(7)
References 211(6)
Index 217
John Stillwell is the author of many books on mathematics; among the best known are Mathematics and its History, Naive Lie Theory, and Elements of Mathematics. He is a member of the inaugural class of Fellows of the American Mathematical Society and winner of the Chauvenet Prize for mathematical exposition.
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