**MATH338**

# Galois Theory

These notes begin with a comprehensive discussion on prime polynomials, with many techniques given for proving that an integer polynomial is prime over **Q**. A particular novelty is the Too Many Primes Test which appears here for the first time.

Then the usual path is followed via splitting fields, automorphisms and Galois groups. But we stick to subfields of the complex numbers. By doing this we avoid some complexities which, for an introduction to the subject, is desirable. We head towards the usual goal of the insolubility of polynomials by radicals. A final chapter gives deals with finite fields.

Many examples of Galois groups worked out in detail. Sometimes students can follow Galois theory without ever being able to compute an actual Galois group, except in the most trivial cases. We try to avoid this happening. Students understand the theory better when they can compute some rather difficult Galois groups.

[Please note that all links are to Adobe .pdf documents. They will open in a separate browser window.]

- Introduction and Contents
- CHAP01 Overview
- CHAP02 Background
- CHAP03 Prime Polynomials
- CHAP04 The Too Many Primes Test
- CHAP05 Field Extensions
- CHAP06 Ruler and Compass Constructions
- CHAP07 Solving a Polynomial Equation
- CHAP08 Galois Groups
- CHAP09 Examples of Galois Groups
- CHAP10 Solubility By Radicals
- CHAP11 Finite Fields
- CHAP12 The Fundamental Theorem of Algebra
- APPENDICES