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Notes for 4H Galois Theory 2002“3

Andrew Baker

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
€€€ hh
N N J ss
€€€ hh ss
€€€ hh ss
€€€ ss
€€€ hh 3
3 hh
Q(ζ3 )
€€€ 3
hh Q
€€€ j
2 jjjjjj
€€€ hhh
€€€ hh
€€€ hh

Gal(E/Q) ∼ S3
nnn zz „„„„2
nnn zzz „„„„
nnn zz „„„„

3 nnnnn 3 zzz
{id, (1 2 3), (1 3 2)}
nnn 3
nnn zz
nnn uu
nnn zz uu
nnn zz uu
nnn  zz
3 uuuu
{id, (2 3)} ˆˆˆ {id, (1 3)} {id, (1 2)} uu
ˆˆˆˆˆ  uu
ˆˆˆˆˆ  uu
ˆˆˆˆˆ  uu
2ˆˆˆˆˆ  
2 u

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

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.


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

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

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-

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.

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

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