文档介绍：Course 311: Hilary Term 2002
Part III: Introduction to Galois Theory
D. R. Wilkins
Contents
3 Introduction to Galois Theory 2
Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2
Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 5
The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 7
Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 7
Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 12
Field Extensions and the Tower Law . . . . . . . . . . . . . . 12
Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 14
Ruler pass Constructions . . . . . . . . . . . . . . . . 16
Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . 24
Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
The Primitive Element Theorem . . . . . . . . . . . . . . . . . 30
The Galois Group of a Field Extension . . . . . . . . . . . . . 31
The Galois correspondence . . . . . . . . . . . . . . . . . . . . 33
Quadratic Polynomials . . . . . . . . . . . . . . . . . . . . . . 35
Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 35
Quartic Polynomials . . . . . . . . . . . . . . . . . . . . . . . 37
The Galois group of the polynomial x4 − 2 . . . . . . . . . . . 38
The Galois group of a polynomial . . . . . . . . . . . . . . . . 39
Solvable polynomials and their Galois groups . . . . . . . . . . 40
A quintic polynomial that is not solvable by radicals . . . . . 44
1
3 Introduction to Galois Theory
Rings and Fields
Definit