A Conversational Introduction to Algebraic Number Theory

Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q. Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, ...

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Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q. Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments.

In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization.

The book is based on the author's notes from a course delivered at the University of Georgia; pains have been taken to preserve the conversational style of the original lectures.

Cover 1
Title page 4
Contents 6
Preface 8
Chapter 1. Getting our feet wet 12
Chapter 2. Cast of characters 20
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Cover 1
Title page 4
Contents 6
Preface 8
Chapter 1. Getting our feet wet 12
Chapter 2. Cast of characters 20
Chapter 3. Quadratic number fields: First steps 30
Chapter 4. Paradise lost —and found 38
Chapter 5. Euclidean quadratic fields 48
Chapter 6. Ideal theory for quadratic fields 60
Chapter 7. Prime ideals in quadratic number rings 70
Chapter 8. Units in quadratic number rings 78
Chapter 9. A touch of class 90
Chapter 10. Measuring the failure of unique factorization 102
Chapter 11. Euler’s prime-producing polynomial and the criterion of Frobenius–Rabinowitsch 116
Chapter 12. Interlude: Lattice points 128
Chapter 13. Back to basics: Starting over with arbitrary number fields 140
Chapter 14. Integral bases: From theory to practice, and back 154
Chapter 15. Ideal theory in general number rings 170
Chapter 16. Finiteness of the class group and the arithmetic of \Z 182
Chapter 17. Prime decomposition in general number rings 190
Chapter 18. Dirichlet’s unit theorem, I 206
Chapter 19. A case study: Units in \Z[√[3]2] and the Diophantine equation 𝑋³-2𝑌³=±1 216
Chapter 20. Dirichlet’s unit theorem, II 226
Chapter 21. More Minkowski magic, with a cameo appearance by Hermite 236
Chapter 22. Dedekind’s discriminant theorem 252
Chapter 23. The quadratic Gauss sum 266
Chapter 24. Ideal density in quadratic number fields 282
Chapter 25. Dirichlet’s class number formula 292
Chapter 26. Three miraculous appearances of quadratic class numbers 306
Index 324
Back Cover 329
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