MA3D5 Resources

Additional Resources:

This page is a copy of for convenience. If anything doesn't work, follow that link instead.


Contact information

lecturer: Marco Streng, office C2.29,
problem class: David Holmes, office B2.01,


Short version: You could use Daan Krammer's lecture notes [Kr], see the detailed timetable below to match the sections of the notes with the lectures. It does not fit completely, so it is good to take notes.

Long version: There are many books on Galois theory. See for example this detailed list of alternative books by Daan Krammer, including a correspondence between chapters in the books and chapters in [Kr]. Another list of books can be found on page 19 of [Re].
We will more or less follow last year's notes by Martin Bright. They have not been typed up, and it is generally advisable to take notes during lectures. It is also advisable to prepare lectures by skimming over the appropriate theory. For that, and for having notes that are not literally the same as what I write on the board, I recommend [Kr], because that is reasonably close to the notes we are using. I will provide references for the lectures in the detailed timetable below, as well as exercises.

[Br] Martin Bright's note on fraction fields and rational function fields
[cy] note on cyclotomic fields
[Kr] Daan Krammer's lecture notes
[Re] Miles Reid's lecture notes
[St] Ian Stewart - Galois Theory, Chapman & Hall, Third Edition 2004.

Assignments and exam

The grade will be based for 15% on 4 homework assignments and for 85% on a final exam. The deadlines for the assignments are:

Besides the assignments, there are also homework excercises that don't count for your grade. Those will be made available via the detailed timetable below.

Assignments will be made available 13 days before the deadline. Part of the problems can be solved with the theory taught up to that moment, and I recommend that you do so. The rest of the problems can be solved with the theory taught at least 6 days before the deadline. For details on which problems can be solved after which lecture, see the detailed timetable below.

Detailed timetable

(references are approximate)

date keywords references exercises
4 Oct
I. Fields and Polynomials
rings, fields, the characterstic of a field, prime fields,
polynomials, rational functions, division with remainder for polynomials
[Kr] 1.1, 2.1, 2.3
[Re] 1.1, 1.2, 2.1, 2.3, [Br])
page 1 of exercises;
all of bonus 1;
1(a,b), 3(d) of assignment 1
7 Oct

euclid's algorithm for polynomials, (unique) factorization of polynomials

[Kr] 2.2
[Re] 2.1 -- 2.3)
half of page 2 of exercises

11 Oct
modular arithmetic for polynomials, fields that are quotient rings of polynomial rings, i.e., polynomial rings modulo ideals
algebraic versus transcendental numbers, the minimal polynomial,
checking irreducibility of polynomials
[Kr] 2.2, 3.1, 3.3
[Re] 2.1, 2.3, 2.4)
page 2 of exercises;
1,4 of assignment 1 (also 3 if you assume 2)
14 Oct
checking irreducibility of polynomials: Gauss' lemma, reduction modulo prime numbers and Eisenstein polynomials,
how to find the minimal polynomial
[Kr] 3.1
[Re] 2.4)

assignment 1;
[Kr] (3.3) - (3.8) (on pages 29,30)

18 Oct
II. Field extensions
field extension, degree, finite and infinite extensions, algebraic and transcendental extensions, the tower law, morphisms, primitive extensions, bijection between roots and morphisms
[Kr] 3.2 - 3.6, 5.1, 5.2
[Re] 3.1)

page 3 of exercises;

1, 2(a),(b), 3(a), 4(a) of assignment 2

Fri 21 Oct

Interlude. Ruler and compass constructions
impossibility to trisect a general angle, constructible field extensions

[Re] 3.2
(or: [St] 7)

and use your notes for the definition of constructible field extensions and constructible points, and what these have to do with each other

3(b) of assignment 2;

or spend more time on page 3 of exercises;

Tues 25 Oct

III. The Galois correspondence
automorphisms, the Galois group, fixed fields, Galois extensions

Interlude. Roots of unity and cyclotomic polynomials

[Kr] 1.6, 4.1 - Proposition 77 and Definition 80
[Re] 1.5 instead of [Kr] 1.6)

assignment 2;
problem 21 of exercises

Fri 28 Oct

how many automorphisms is "enough"?
examples of Galois and non-Galois extensions

[Kr] Proposition 78 without its proof (mistake in proof)

assignment 2;
problem 21 again, or [Kr] (4.17)

Tues 1 Nov

Artin's lemma
IV. The fundamental theorem of Galois theory

the fundamental theorem, stable intermediate fields, normal field extensions, characterizations of Galois subextensions
example: Q(zeta_5)/Q

[Re] Proposition 4.8 together with 4 of assignment 3


[Kr] 4.4 - 4.6, 5.3


either [Kr] exercise (5.6) or [Re] Claim 4.19

22, 23 of exercises;

4 of assignment 3

Fri 4 Nov continuation of example,
and another example: Q(4th root of 2, i)
[St] 13
[Re] 4.22)

24 of exercises;

1, 4 of assignment 3

Tues 8 Nov

continuation of example
V. Splitting fields
existence of splitting fields, isomorphism extension theorem

[Kr] 6.1:
page 47

assignment 3;

25 of exercises

Fri 11 Nov normal extensions versus splitting fields, Galois groups of splitting fields as permutation groups of roots, normal closure [Kr] 6.2, 8.1

assignment 3;

26 and 27 of exercises

Tues 15 Nov

continuation of normal closure, remark on algebraically closed fields
VI. Separability
repeated roots and derivatives, definition of separable polynomials and extensions, characterisation of inseparable polynomials, perfect fields, Galois if and only if normal and separable (statement)

[Kr] 8.1, remainder of Chapter 6
1 of assignment 4; 28 and 29 of exercises
Fri 18 Nov

Galois if and only if normal and separable (proof), statement without proof of the primitive element theorem
VII. Finite Fields

structure of the multiplicative group, Frobenius automorphism, order q is a power of a prime

[Kr] Chapter 6 (again) and Chapter 7 (for finite fields)
1 and 2 of assignment 4;
30 of exercises;
(7.3) of [Kr]
Tues 22 Nov

the splitting field of xq-x, existence and uniqueness, Galois groups of finite fields
VIII. Cyclotomic extensions
definition and Galois group, example, irreducibility of cyclotomic polynomials

[Kr] Chapter 7 (finite fields)


[cy] (cyclotomic fields)

1, 2, 3 of assignment 4;
[Kr] (7.2), (7.4)--(7.7)
Fri 25 Nov abelian extensions, statement of the Kronecker-Weber theorem,(in)constructibility of a regular n-gon
IX. Solvability in radicals
definition of a solvable extension, and what this has to do with roots
[cy] and [Kr] 1.3, 8.3
assignment 4 ;

31 and 34 of exercises
Tues 29 Nov normal closure of a solvable extension, definition of a solvable polynomial, lemma on cyclic extensions, solvable groups, examples of solvable groups, solvable polynomials imply solvable groups
[Kr] 1.3, 8.2, 8.3 32, 33, 34 of exercises
Fri 2 Dec

statement without proof of the converse, S5 is not solvable, a sufficient criterion for the Galois group to be S5.

[Kr] 8.3
35 and 36 of exercises
Tues 6 Dec example of a non-solvable polynomial, insolvability of the general quintic!
X. Calculating Galois groups
elementary symmetric polynomials, discriminants, odd and even permutations of the roots, transitive groups, cubic polynomials, explicitly solving cubics in radicals
[Kr] 1.4 (symmetric polynomials), 1.7 (solving the cubic) and take notes (discriminant, odd and even permutations versus discriminant, Galois group of a cubic) 37 -- 39 of exercises
Fri 9 Dec

XI. Quadratic reciprocity
(does not belong to the syllabus itself, but is good practice in techniques and theory from the module)
quadratic fields of Q(zetap), proof of quadratic reciprocity using Galois theory

take notes

40 -- 42 of exercises
(which actually belongs to the previous lecture)


Everything mentioned above, except the last lecture

everything mentioned above, plus old exams and their solutions

Exercise sheets:

Other links