Andrew Baker

[05/02/2003]

Department of Mathematics, University of Glasgow.

E-mail address: a.baker@maths.gla.ac.uk

URL: http://www.maths.gla.ac.uk/∼ajb

√

Q( 3 2, ζ3 )

ggg I ss

ggggg mm ss

ggggg mmmmm ss

ggggg 2

m ss

ggggg mmm 2 ss

ggggg 2 m

√ √ √2 ss 3

ggg ss

Q( 3 2) 3 3

Q( 2 ζ3 ) Q( 2 ζ3 ) ss

ss

hh

N N J ss

hh ss

hh ss

ss

hh

hh 3

3 hh

Q(ζ3 )

3

hh

jjj

hh Q

j

2 jjjjjj

hhh

jj

hh

jjjj

hh

jjj

jjjj

QS

Gal(E/Q) ∼ S3

=

„„„„

nnn zz „„„„2

nnn zzz „„„„

nnn zz „„„„

nnn

z

3 nnnnn 3 zzz

{id, (1 2 3), (1 3 2)}

nnn 3

z

zz

nnn zz

nnn uu

nnn zz uu

uu

nnn zz uu

nnn zz

u

3 uuuu

{id, (2 3)} {id, (1 3)} {id, (1 2)} uu

uu

uu

uu

uu

2

uu

2

2 u

{id}

√

The Galois Correspondence for Q( 3 2, ζ3 )/Q

ii

Introduction: What is Galois Theory?

Much of early algebra centred around the search for explicit formul¦ for roots of polynomial

equations in one or more unknowns. The solution of linear and quadratic equations in one

unknown was well understood in antiquity, while formul¦ for the roots of general real cubics and

quartics was solved by the 16th century. These solutions involved complex numbers rather than

just real numbers. By the early 19th century no general solution of a general polynomial equation

˜by radicals™ (i.e., by repeatedly taking n-th roots for various n) was found despite considerable

e¬ort by many outstanding mathematicians. Eventually, the work of Abel and Galois led to a

satisfactory framework for fully understanding this problem and the realization that the general

polynomial equation of degree at least 5 could not always be solved by radicals. At a more

profound level, the algebraic structure of Galois extensions is mirrored in the subgroups of their

Galois groups, which allows the application of group theoretic ideas to the study of ¬elds. This

Galois Correspondence is a powerful idea which can be generalized to apply to such diverse

topics as ring theory, algebraic number theory, algebraic geometry, di¬erential equations and

algebraic topology. Because of this, Galois theory in its many manifestations is a central topic

in modern mathematics.

In this course we will focus on the following topics.

• The solution of polynomial equations over a ¬eld, including relationships between roots,

methods of solutions and location of roots.

• The structure of ¬nite and algebraic extensions of ¬elds and their automorphisms.

We will study these in detail, building up a theory of algebraic extensions of ¬elds and their

automorphism groups and applying it to questions about roots of polynomial equations. The

techniques we discuss can also be applied to the following topics some of which will be met by

students taking advanced degrees.

• Classical topics such as ˜squaring the circle™, ˜duplication of the cube™, constructible

numbers and constructible polygons.

• Applications of Galois theoretic ideas in Number Theory, the study of di¬erential

equations and Algebraic Geometry.

There are many good introductory books on Galois Theory and we list some of them in the

bibliography at the end.

Suggestions on using these notes

These notes cover more than the content of the course and should be used in parallel with

the lectures. The problem sets contain samples of the kind of problems likely to occur in the

¬nal examination and should be attempted as an important part of the learning process.

™

The symbol means the adjacent portion of the notes is not examinable.

™™¦

™

Contents

Introduction: What is Galois Theory? ii

Suggestions on using these notes ii

Chapter 1. Integral domains, ¬elds and polynomial rings 1

1.1. Recollections on integral domains and ¬elds 1

1.2. Polynomial rings 5

1.3. Identifying irreducible polynomials 10

1.4. Finding roots of complex polynomials of small degree 13

1.5. Automorphisms of rings and ¬elds 16

Exercises on Chapter 1 20

Chapter 2. Fields and their extensions 23

2.1. Fields and sub¬elds 23

2.2. Simple and ¬nitely generated extensions 25

Exercises on Chapter 2 28

Chapter 3. Algebraic extensions of ¬elds 29

3.1. Algebraic extensions 29

3.2. Splitting ¬elds and Kronecker™s Theorem 32

3.3. Monomorphisms between extensions 35

3.4. Algebraic closures 37

3.5. Multiplicity of roots and separability 40

3.6. The Primitive Element Theorem 43

3.7. Normal extensions and splitting ¬elds 45

Exercises on Chapter 3 45

Chapter 4. Galois extensions and the Galois Correspondence 47

4.1. Galois extensions 47

4.2. Working with Galois groups 47

4.3. Subgroups of Galois groups and their ¬xed ¬elds 49

4.4. Sub¬elds of Galois extensions and relative Galois groups 50

4.5. The Galois Correspondence 51

4.6. Galois extensions inside the complex numbers and complex conjugation 53

4.7. Kaplansky™s Theorem 54

Exercises on Chapter 4 56

Chapter 5. Galois extensions for ¬elds of positive characteristic 59

5.1. Finite ¬elds 59

5.2. Galois groups of ¬nite ¬elds and Frobenius mappings 62

5.3. The trace and norm mappings 64

Exercises on Chapter 5 65

Chapter 6. A Galois Miscellany 67

6.1. A proof of the Fundamental Theorem of Algebra 67

6.2. Cyclotomic extensions 67

6.3. Artin™s Theorem on linear independence of characters 71

6.4. Simple radical extensions 73

iii

iv CONTENTS

6.5. Solvability and radical extensions 74

6.6. Galois groups of even and odd permutations 77

6.7. Symmetric functions 79

Exercises on Chapter 6 80

Bibliography 83

Solutions 85

Chapter 1 85

Chapter 2 91

Chapter 3 93

Chapter 4 95

Chapter 5 97

Chapter 6 98

CHAPTER 1

Integral domains, ¬elds and polynomial rings

In these notes, a ring will always be a ring with unity 1 which is assumed to be non-zero,

1 = 0. In practice, most of the rings encountered will also be commutative. An ideal I R

always means a 2-sided ideal. An ideal I R in a ring R is proper if I = R, or equivalently if

I R. For a ring homomorphism • : R ’’ S, the unity of R is sent to that of S, i.e., •(1) = 1.

Definition 1.1. Let • : R ’’ S be a ring homomorphism.

• • is a monomorphism if it is injective.

• • is an epimorphism if it is surjective.

• • is an isomorphism if it is both a monomorphism and an epimorphism.

Remark 1.2. The following are equivalent formulations of the notions in De¬nition 1.1.

• • is a monomorphism if and only if for r1 , r2 ∈ R, if •(r1 ) = •(r2 ) then r1 = r2 , or

equivalently, if r ∈ R with •(r) = 0 then r = 0.

• • is an epimorphism if and only if for every s ∈ S there is an r ∈ R with •(r) = s.

• • is an isomorphism if and only if it is invertible (and whose inverse is also an isomor-

phism).

1.1. Recollections on integral domains and ¬elds

The material in this section is standard and most of it should be familiar. For details

see [3, 4] or any other book containing introductory ring theory.

Definition 1.3. A commutative ring R in which there are no zero-divisors is called an

integral domain or an entire ring. This means that for u, v ∈ R,

uv = 0 =’ u = 0 or v = 0.

Example 1.4. The following rings are integral domains.

(i) The ring of integers, Z.

(ii) If p is a prime, the ring of integers modulo p, Fp = Z/p = Z/(p).

(iii) The rings of rational numbers, Q, real numbers, R, and complex numbers, C.

(iv) The polynomial rings Z[X], Q[X], R[X] and C[X].

Definition 1.5. Let I R be a proper ideal in a commutative ring R.

• I is a prime ideal if for u, v ∈ R,

uv ∈ I =’ u ∈ I or v ∈ I.

Similarly, I is a maximal ideal R if whenever J R is a proper ideal and I ⊆ J then

J = I.

• I R is principal if

I = (p) = {rp : r ∈ R}

for some p ∈ R. Notice that if p, q ∈ R, then (q) = (p) if and only if q = up for some

unit u ∈ R. We also write p | x if x ∈ (p).

• p ∈ R is prime if (p) R is a prime ideal; this is equivalent to the requirement that

whenever p | xy with x, y ∈ R then p | x or p | y.

• R is a principal ideal domain if it is an integral domain and every ideal I R is principal.

The following fundamental example should be familiar.

1

2 1. INTEGRAL DOMAINS, FIELDS AND POLYNOMIAL RINGS

Example 1.6. Every ideal I Z is principal, so I = (n) for some n ∈ Z which we can always

take to be non-negative, n 0. Hence Z is a principal ideal domain.

Proposition 1.7. Let R be a commutative ring and I R an ideal.

(i) The quotient ring R/I is an integral domain if and only if I is a prime ideal.

(ii) The quotient ring R/I is a ¬eld if and only if I is a maximal ideal.

Example 1.8. If n 0, the quotient ring Z/n = Z/(n) is an integral domain if and only if

n is a prime.

For any (not necessarily commutative) ring with unit there is an important ring homomor-

phism · : Z ’’ R called the unit or characteristic homomorphism which is de¬ned by

±

1 + · · · + 1 if n > 0,

n

·(n) = n1 = ’(1 + · · · + 1) if n < 0,

’n

0 if n = 0.

Since the unit 1 ∈ R is non-zero, ker · Z is a non-zero ideal and using the Isomorphism Theorems

we see that there is a quotient monomorphism · : Z/ ker · ’’ R which allows us to identify the