Neukirch - Algebraic Number Theory

Grundlehren Der Mathematischen Wissenschaften 322

Algebraic Number Theory

"The desire to present number theory as much as possible from a unified theoretical point of view seems imperative today, as a result of the revolutionary development that number theory has undergone in the last decades in conjunction with ‘arithmetic algebraic geometry’. The immense success that this new geometric perspective has brought about - for instance, in the context of the Weil conjectures, the Mordell conjecture, of problems related to the conjectures of Birch and Swinnerton-Dyer - is largely based on the unconditional and universal application of the conceptual approach."

Chapter 1: Algebraic Integers Edit

Notes for Chapter 1 sections 2-8:

Section 1.1: The Gaussian Integers Edit

  • Proof of Theorem 1.1: Thus we have $ p | x^2 + 1 $... What's meant here is that p divides the number $ (2n)!^2 + 1 $, not the the polynomial $ x^2+1 $ is divisible by p. Incidentally, this proof is due to Dedekind.

Section 1.2: Integrality Edit

Section 1.3: Ideals Edit

Section 1.4: Lattices Edit

Section 1.5: Minkowski Theory Edit

Section 1.6: The Class Number Edit

Section 1.7: Dirichlet's Unit Theorem Edit

Section 1.8: Extensions of Dedekind Domains Edit

Section 1.9: Hilbert's Ramification Theory Edit

Section 1.10: Cyclotomic Fields Edit

Section 1.11: Localization Edit

Section 1.12: Orders Edit

Section 1.13: One-Dimensional Schemes Edit

Section 1.14: Function Fields Edit

Chapter 2: The Theory of Valuations Edit

Chapter 3: Riemann-Roch Theory Edit

Chapter 4: Abstract Class Field Theory Edit

Chapter 5: Local Class Field Theory Edit

Chapter 6: Global Class Field Theory Edit

Chapter 7: Zeta Functions and L-Series Edit