Group Theory and Galois Extension

Abstract

Abstract Galois Theory is an important branch of algebra. It establishes corresponding contacts between groups and fields. In this article, we mainly discuss some basic field extensions and relate them with Galois groups. At the same time, we also explore the structure of radical roots. And we use Sylow’s Theorem in some easy field extensions to check middle fields. Finally, we mention the Sylow’s Theorem in finite field. Our main tools are basic polynomial theory, group theory and field theory.