Teoria de galois

Galois Theory
Dr P.M.H. Wilson1 Michaelmas Term 2000

1 A LT

EXed by James Lingard — please send all comments and corrections to james@lingard.com

These notes are based on a course of lectures given by Dr Wilson during Michaelmas Term 2000 for Part IIB of the Cambridge University Mathematics Tripos. In general the notes follow Dr Wilson’s lectures very closely, although there arecertain changes. In particular, the organisation of Chapter 1 is somewhat different to how this part of the course was lectured, and I have also consistently avoided the use of a lower-case k to refer to a field — in these notes fields are always denoted by upper-case roman letters. These notes have not been checked by Dr Wilson and should not be regarded as official notes for the course. In particular, theresponsibility for any errors is mine — please email me at james@lingard.com with any comments or corrections. James Lingard October 2001

Contents
1 Revision from Groups, Rings and Fields 1.1 1.2 1.3 1.4 1.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of simple algebraic extensions . . . . . . . . . . . . . . . . . . . . . Tests forirreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The degree of an extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 3 3 4 5 7 7 8 10 11 12 12 12 16 16 17 19 20 21 24 24 24 27 30 30 32 33 34

2 Separability 2.1 2.2 2.3 2.4 Separable polynomials andformal differentiation . . . . . . . . . . . . . . . . . . Separable extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Primitive Element Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . Trace and norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Algebraic Closures 3.1 3.2 Definitions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Existence and uniqueness of algebraic closures . . . . . . . . . . . . . . . . . . . .

4 Normal Extensions and Galois Extensions 4.1 4.2 4.3 4.4 4.5 Normal extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed fields and Galoisextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Galois correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Galois groups of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Galois Theory of Finite Fields 5.1 5.2 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galois groups of finiteextensions of finite fields . . . . . . . . . . . . . . . . . . .

6 Cyclotomic Extensions 7 Kummer Theory and Solving by Radicals 7.1 7.2 7.3 7.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . Insolubility of the general quintic by radicals . . . . . . . . . . . . . . . . . . . .

1

1
1.1

Revision from Groups, Rings and Fields
Field extensions

Suppose K and L are fields. Recall that a non-zero ring homomorphism θ : K → L is necessarily injective (since ker θ ¡ K and so ker θ = {0}) and satisfies θ(a/b) = θ(a)/θ(b). Therefore θ is a homomorphism of fields.Definition A field extension of K is given by a field L and a non-zero homomorphism θ : K → L. Such a θ will also be called an embedding of K into L. Remark In fact, we often identify K with its image θ(K) ⊆ L, since θ : K → θ(K) is an isomorphism, and denote the extension by L/K or K → L. Lemma 1.1 If {Ki }i∈I is any collection of subfields of a field L, then Proof Easy exercise from the axioms. Definition...