Galois Theory, Algebraic Number Theory, and Zeta Functions

Abstract

The goal of these lectures is to explain the fundamentals of Galois theory and algebraic number theory and to get the reader to the point that he or she can make routine calculations. The following material has little in the way of prerequisites. Many examples will be given along the way and sometimes the examples will serve instead of formal proofs. In the last Section, we introduce the zeta functions of algebraic number fields. These functions can be factored into products of L-functions according to representations of Galois groups. Especially here, we will proceed by example. These lectures have been abstracted from my long promised forthcoming book [2]. A complete treatment of any of the topics here is somewhat beyond the length requirements of these lectures although we will come surprisingly close in some instances. I would also recommend Hecke’s book [1] which has been a classic for almost 70 years.