Algebraic Number Theory, a Computational Approach William A. Stein Contents Preface 1 Introduction 1.1 Mathematical background 1.2 What is algebraic number theory? 1.2.1 Topics in this book 1.3 Some applications of algebraic number theory 2 Basic Commutative Algebra 2.1 Finitely Generated Abelian Groups 2.2 Noetherian Rings and Modules 2.2.1 The Ring ℤ\mathbb{Z} is Noetherian 2.3 Rings of Algebraic Integers 2.3.1 Minimal Polynomials 2.3.2 Number fields, rings of integers, and orders 2.3.3 Function fields 2.4 Norms and Traces 2.5 Recognizing Algebraic Numbers using LLL 2.5.1 LLL Reduced Basis 2.5.2 What LLL really means 2.5.3 Applying LLL 3 Unique Factorization of Ideals 3.1 Dedekind Domains 3.2 Factorization of Ideals 4 Factoring Primes 4.1 The Problem 4.1.1 Geometric Intuition 4.1.2 Examples 4.2 A Method for Factoring Primes that Often Works 4.3 A General Method 4.3.1 Inessential Discriminant Divisors 4.3.2 Remarks on Ideal Factorization in General 4.3.3 Finding a pp-Maximal Order 4.3.4 General Factorization Algorithm of Buchman-Lenstra 5 The Chinese Remainder Theorem 5.1 The Chinese Remainder Theorem 5.1.1 CRT in the Integers 5.1.2 CRT in General 5.2 Structural Applications of the CRT 5.3 Computing Using the CRT 5.3.1 Sage 5.3.2 5.3.3 PARI 6 Discriminants and Norms 6.1 Viewing 𝒪K\mathcal{O}_{K} as a Lattice in a Real Vector Space 6.1.1 A Determinant 6.2 Discriminants 6.3 Norms of Ideals 7 Finiteness of the Class Group 7.1 The Class Group 7.2 Class Number 1 7.3 More About Computing Class Groups 8 Dirichlet’s Unit Theorem 8.1 The Group of Units 8.2 Examples with Sage 8.2.1 Pell’s Equation 8.2.2 Examples with Various Signatures 9 Decomposition and Inertia Groups 9.1 Galois Extensions 9.2 Decomposition of Primes: efg=nefg=n 9.2.1 Special Cases 9.2.2 Definitions and Terminology 9.3 The Decomposition Group 9.3.1 Galois groups of finite fields 9.3.2 The Exact Sequence 9.4 Frobenius Elements 9.5 The Artin Conjecture 10 Elliptic Curves and LL-functions 10.1 Groups Attached to Elliptic Curves 10.1.1 Abelian Groups Attached to Elliptic Curves 10.1.2 A Formula for Adding Points 10.1.3 Other Groups 10.2 Galois Representations 10.2.1 Modularity of Elliptic Curves over ℚ\mathbb{Q} 11 Galois Cohomology 11.1 Group Rings and Modules 11.2 Group Cohomology 11.2.1 The Main Theorem 11.2.2 Example Application of the Theorem 11.3 Inflation and Restriction 11.4 Galois Cohomology 12 The Weak Mordell-Weil Theorem 12.1 Kummer Theory of Number Fields 12.2 Proof of the Weak Mordell-Weil Theorem Bibliography