The Artin-Schreier Theorem in Galois Theory Open Access

Cheng, Yining (2017)

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We first list and state some basic definitions and theorems of the Galois theory of finite extensions, as well as state and prove the Kummer theory and the Artin-Schreier extensions as prerequisites. The main part of this thesis is the proof of the Artin-Schreier Theorem, which states that an algebraic closed field having finite extension with its subfield F has degree at most two and F must have characteristic 0. After the proof, we will discuss the applications for the Artin-Schreier Theorem.

Table of Contents


1 Introduction 1

2 Basic Definitions, Theorems, And Some Lemmas 2

  1. 2.1 Field Theory ................................ 2
  2. 2.2 Galois Theory................................ 11

3 The Artin-Schreier Extension 21

4 Kummer Theory 26

5 The Artin-Schreier Theorem 28

6 Applications 33

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