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

Notes for Chapter 1 sections 2-8: https://themodularperspective.com/2019/03/25/graduate-text-notes-current-as-of-3-25-2019/

Section 1.1: The Gaussian Integers

• 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.