Galois Theory, TCD 2017/18

Summary of classes

Lecture	Topics covered	Lecture notes/slides	Homeworks/Tutorials/Solutions
1 (26/09)	Introduction and motivation for Galois theory. Solving the cubic.
2 (28/09)	Solving the quartic. Main theorem on symmetric polynomials.
3 (29/09)	Recollections from group theory: group of small orders, group actions, a group whose order is a prime power has a nontrivial centre, all orders of elements in a finite Abelian group divide the largest one.
4 (03/10)	A finite subgroup of K^×, K a field, is cyclic. Classification of finite fields: characteristic, the set of roots of x^q-x in any field extension of Z/pZ is a field. Reduction to the existence and uniqueness of splitting fields.
5 (05/10)	Definition of a splitting field; statement of the theorem on their existence and uniqueness. Algebraic and transcendental elements. An element a is algebraic over k iff k(a)=k[a].
6 (06/10)	Adjoining a root of an irreducible polynomial. Degree of an extension, tower law. Existence and uniqueness of splitting fields.		[HW1]
7 (10/10)	Normal and separable extensions. Example of a non-normal extension. Normal extensions are splitting fields. Separable extensions: characteristic 0 and characteristic p.
8 (12/10)	Example of a non-separable extension: splitting field of x^p-a over a field of characteristic p. The automorphism group of a field. Examples. Real numbers have no nontrivial automorphisms.
9 (13/10)	Galois group of an extension. The number of elements of the Galois group of a finite extension does not exceed the degree.
10 (17/10)	Galois extensions (finite normal separable extensions K:k) are invariants of finite subgroups of the automorphism group Aut(K).		[HW1 solutions]
11 (19/10)	Galois correspondence: fields between k and K are in one-to-one correspondence with subgroups of the Galois group Gal(K:k). Normality of extensions and subgroups.		[HW2]
12 (20/10)	Examples for Galois correspondence (integrality problem from a maths olympiad, the field Q(√2,√3), the field of fifth roots of unity).
13 (24/10)	The n-th cyclotomic polynomial, its integrality and irreducibility.
14 (26/10)	Degree of the cyclotomic field is the Euler function. The Galois group of the n-th cyclotomic field is (Z/nZ)^×. Constructibility of the regular n-gon with ruler and compass and Fermat primes: the elementary step.
15 (27/10)	Constructibility of the regular n-gon with ruler and compass and Fermat primes: the Galois theory step. Example of the 17-th cyclotomic field.
16 (31/10)	Galois theory, Sylow 2-subgroups, and the Fundamental Theorem of Algebra. The Galois group of a generic polynomial of degree n is S_n.		[HW2 solutions]
17 (02/11)	Solvable groups. Examples. Subgroups and quotient groups of solvable groups are solvable. The converse statement. Three equivalent definition of solvable groups.		[HW3]
18 (03/11)	The group A₅ is not solvable (in fact, is simple), and the impact of this statement for solving equations in radicals. Radical and solvable extensions: examples. For a field with enough roots of unity, the Galois group is solvable if and only if the extension is radical: reduction to the case of cyclic extensions.
	Reading week, no classes.
19 (14/11)	For a field with enough roots of unity, the Galois group is solvable if and only if the extension is radical: the case of cyclic extensions. Linear independence of homomorphisms.		[HW3 solutions]
20 (16/11)	Adjoining roots of unity does not change the solvability of the Galois group. Normal closure does not change the radical property. Main theorem: an extension is solvable if and only if the Galois group of its normal closure is solvable.		[HW4]
21 (17/11)	Computing and using Galois groups. A transitive subgroup of S₅ containing a transposition coincides with S₅, and how it implies that x⁵-6x+3 is not solvable in radicals. A transitive subgroup of S_n containing a transposition and an (n-1)-cycle coincides with S_n. Statement of Dedekind's theorem on cycle types of Galois group arising from reduction modulo p.
22 (21/11)	Dedekind's theorem on cycle types of Galois group arising from reduction modulo p. Kronecker's theorem on computing Galois groups.
23 (23/11)	Primitive element theorem. There exist irreducible polynomials over F_p of all possible degrees. Example of a polynomial with Galois group S_n over Q.
24 (24/11)	Solving the cubic and the quartic using Galois theory. Discriminant of a polynomial and its Galois-theoretic meaning. Galois groups of quartics: distinguishing between S₄, A₄, K₄, and (D₄ or Z/4Z).
25 (28/11)	Galois groups of quartics: distinguishing between D₄ and Z/4Z (irreducibility test, Kappe-Warren test).		[HW4 solutions] [HW5 (optional)]
26 (30/11)	Inverse Galois problem. Realisation of groups of small orders, including the quaternion group Q₈. Realisability of solvable groups: statement of Shafarevich theorem.
27 (01/12)	Realisation of Abelian groups. Statement of Kronecker-Weber theorem. The normal basis theorem.	[Draft notes (PDF)]
28 (05/12)	Discussion of selected homework questions.
29 (07/12)	Discussion of selected homework questions. Revision of the module.
30 (08/12)	No classes on Friday 08/12.

About this module

Galois theory demonstrates how to use symmetries of objects to learn something new about properties of those objects, on the example of polynomial equations in one variable with coefficients in a field, and specifically roots of those equations. Students taking the module will see how basics of group theory can be used for solving problems outside group theory, in particular for proving a celebrated result of Abel on non-existence of formulas to solve equations of degree 5 using only arithmetic operations and extracting roots.

Syllabus

Recollection of relevant results in on groups, fields, and rings. Polynomial rings: UFD / PID property, Gauss lemma, Eisenstein's criterion.
Algebraic field extensions. Tower Law, ruler and compass constructions.
Splitting fields, and their properties. Classification of finite fields.
Normal and separable extensions. The Primitive Element Theorem. Galois extensions. The Galois correspondence.
Algorithm for computing the Galois group of a given polynomial. Specific computations of Galois groups of polynomials of low degree.
Solubility by radicals. Cyclic, Abelian, solvable field extensions. Abel's theorem on equations of degree five.
Abelian and cyclotomic extensions. Towards the Kronecker-Weber Theorem.

Assessment

There will be several home assignments (roughly every 2-3 weeks) that contribute 20% of your final mark. The rest of the mark (80%) comes from the final exam.

Disclaimer

The person who is solely responsible for the choice of content on this page is Vladimir Dotsenko. Any views expressed here do not necessarily represent the official views of Trinity College Dublin.