RINGS AND GALOIS THEORY

Contents
1Rings6
Definition, examples, elementary properties6
Units and zero divisors8
Ring constructions11
Polynomial rings12
Product of rings13
Quotient fields.13
Rings of functions.14
Non-commutative rings14
1.4.6Group rings (optional)17
Exercises17
Exercises in non-commutative rings20
2Ideals and ring homomorphisms23
Ring homomorphisms23
2.1.2Examples.23
The factor ring R/I27
Ideal arithmetic29
Exercises33
3Polynomials, unique factorisation37
Polynomials37
Irreducible elements40
Unique factorisation42
Euclidean rings44
Exercises45
4Factorisation of polynomials47
Polynomial factorisation in practice47
Gauss Lemma47
Factorization over a unique factorization domain49
Unique factorisation in polynomial rings50
Eisenstein’s criterion51
Exercises52
25Prime and maximal ideals55
Prime ideals55
Maximal ideals57
Existence of maximal ideals (optional)59
Zorn’s Lemma (optional)61
Exercises62
6Fields66
Prime fields and characteristic66
Algebraic and transcendental elements67
Finite and algebraic extensions70
Composite extensions72
Determination of minimal polynomials74
Exercises75
7Motivatie Galoistheorie (in Dutch)79
Inleiding80
Construeerbaarheid84
8Splitting fields and Galois groups90
Splitting fields90
The Galois group91
Galois extensions94
Exercises98
9The Main Theorem of Galois theory99
Fixed fields99
Main Theorem101
Examples of Galois correspondences102
Simple extensions106
Exercises107
10Solving equations111
Symmetric polynomials111
Solution of the cubic equation111
Solution of the quartic equation112
Radical extensions113
Solvability115
Main Theorem117
Exercises119
11Finite fields121
Existence, unicity121
Exercises122