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1. Extensions of Fields 1

1.1. Definitions 1

1.2. The characteristic of a field 1

1.3. The polynomial ring F[X] 2

1.4. Factoring polynomials 2

1.5. Extension fields;de grees 4

1.6. Construction of some extensions 4

1.7. Generators of extension fields 5

1.8. Algebraic and transcendental elements 6

1.9. Transcendental numbers 8

1.10. Constructions with straight-edge and compass. 9

2. Splitting Fields;A lgebraic Closures 12

2.1. Maps from simple extensions. 12

2.2. Splitting fields 13

2.3. Algebraic closures 14

3. The Fundamental Theorem of Galois Theory 18

3.1. Multiple roots 18

3.2. Groups of automorphisms of fields 19

3.3. Separable, normal, and Galois extensions 21

3.4. The fundamental theorem of Galois theory 23

3.5. Constructible numbers revisited 26

3.6. Galois group of a polynomial 26

3.7. Solvability of equations 27

Copyright 1996 J.S. Milne. You may make one copy of these notes for your own personal use.

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ii J.S. MILNE

4. Computing Galois Groups. 28

4.1. When is Gf ⊂ An? 28

4.2. When is Gf transitive? 29

4.3. Polynomials of degree ≤3 29

4.4. Quartic polynomials 29

4.5. Examples of polynomials with Sp as Galois group over Q 31

4.6. Finite fields 32

4.7. Computing Galois groups over Q 33

5. Applications of Galois Theory 36

5.1. Primitive element theorem. 36

5.2. Fundamental Theorem of Algebra 38

5.3. Cyclotomic extensions 39

5.4. Independence of characters 41

5.5. Hilbert’s Theorem 90. 42

5.6. Cyclic extensions. 44

5.7. Proof of Galois’s solvability theorem 45

5.8. The general polynomial of degree n 46

Symmetric polynomials 46

The general polynomial 47

A brief history 49

5.9. Norms and traces 49

5.10. Infinite Galois extensions (sketch) 52

6. Transcendental Extensions 54

FIELDS AND GALOIS THEORY 1

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