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# A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond $\mathbb {Z}$

### About this Title

**Paul Pollack**, *University of Georgia, Athens, GA*

Publication: The Student Mathematical Library

Publication Year:
2017; Volume 84

ISBNs: 978-1-4704-3653-7 (print); 978-1-4704-4125-8 (online)

DOI: https://doi.org/10.1090/stml/084

MathSciNet review: MR3675899

MSC: Primary 11-01; Secondary 11Rxx

### Table of Contents

**Front/Back Matter**

**Chapters**

- Getting our feet wet
- Cast of characters
- Quadratic number fields: First steps
- Paradise lost — and found
- Euclidean quadratic fields
- Ideal theory for quadratic fields
- Prime ideals in quadratic number rings
- Units in quadratic number rings
- A touch of class
- Measuring the failure of unique factorization
- Euler’s prime-producing polynomial and the criterion of Frobenius–Rabinowitsch
- Interlude: Lattice points
- Back to basics: Starting over with arbitrary number fields
- Integral bases: From theory to practice, and back
- Ideal theory in general number rings
- Finiteness of the class group and the arithmetic of $\overline {\mathbb {Z}}$
- Prime decomposition in general number rings
- Dirichlet’s unit theorem, I
- A case study: Units in $\mathbb {Z}[\sqrt [3]{2}]$ and the Diophantine equation $X^3-2Y^3=\pm 1$
- Dirichlet’s unit theorem, II
- More Minkowski magic, with a cameo appearance by Hermite
- Dedekind’s discriminant theorem
- The quadratic Gauss sum
- Ideal density in quadratic number fields
- Dirichlet’s class number formula
- Three miraculous appearances of quadratic class numbers