Absolute Galois group as a profinite group

Abstract

In this paper we will discuss the absolute Galois group, the Galois group of Q where Q is an algebraic closure of Q. We will begin with a discussion of Galois groups and Galois theory and why they are important. Then we will form a better understanding of what a profinite group looks like by examining the p-adic integers Zp. In particular we will prove several properties for profinite groups as a whole so that we can then apply those properties to the absolute Galois group. Finally we will apply the structure and topology we learned for profinite groups to form the absolute Galois group, while discussing the differences from the p-adic integers and the complications that arise. Included in this discussion will be a somewhat unorthodox proof of the uncountability of the absolute Galois group involving compactness and some basic Galois theory applied to the splitting fields of x^2 - p for all primes p.