# Part II - Galois Theory

## Lectured by C. Birkar, Michaelmas 2015

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# Contents

- V Full version
- 0 Introduction
- 1 Solving equations
- 2 Field extensions
- 2.1 Field extensions
- 2.2 Ruler and compass constructions
- 2.3 K-homomorphisms and the Galois Group
- 2.4 Splitting fields
- 2.5 Algebraic closures
- 2.6 Separable extensions
- 2.7 Normal extensions
- 2.8 The fundamental theorem of Galois theory
- 2.9 Finite fields
- 3 Solutions to polynomial equations
- 3.1 Cyclotomic extensions
- 3.2 Kummer extensions
- 3.3 Radical extensions
- 3.4 Solubility of groups, extensions and polynomials
- 3.5 Insolubility of general equations of degree 5 or more
- 4 Computational techniques