Galois Theory, TCD 2015/16

Notes

Summary of classes

Lecture	Topics covered	Lecture notes/slides	Homeworks/Tutorials/Solutions
1 (29/09)	Motivation and intuition for Galois theory. Solving the cubic.
2 (29/09)	Solving the quartic. Main theorem on symmetric polynomials (statement).
3 (01/10)	Main theorem on symmetric polynomials (proof). Background theoretical material on groups. Finite groups of small orders.
4 (06/10)	Background theoretical material on groups. Class formula. The group whose order is a prime power has a nontrivial centre. The quotient of a noncommutative group by its centre is non-cyclic.
5 (06/10)	Background theoretical material on fields and rings. Integral domains. Polynomials. Fields of fractions. Prime and maximal ideals. Adjoining the root of an irreducible polynomial.
6 (08/10)	Existence and uniqueness of field extensions. Algebraic and transcendental elements. Equality k[a]=k(a) for an algebraic element a. Characteristic of a field. Finite fields.
7 (13/10)	Background on field extensions. Tower law. Finite fields. The number of elements in a finite field is a prime power. Uniqueness theorem for finite fields.
8 (13/10)	Extending a subfield inclusion to a splitting field. Uniqueness of splitting fields. Normal extensions.
9 (15/10)	Splitting fields are normal extensions of finite degree. Separable extensions.
10 (20/10)	Galois group of a field extension. Number of automorphisms, degreee, and normality/separability.
11 (20/10)	Galois extensions (finite-normal-separable) are fixed fields of finite subgroups of the automorphism group.
12 (22/10)	Main theorem of Galois theory: Galois correspondence and its properties.
13 (27/10)	Example of the field extension Q(√2,√3) and Q(ζ) with ζ a primitive 5th root of 1.
14 (27/10)	Cyclotomic fields. Integrality and irreducibility of the cyclotomic polynomial. Galois group of the cyclotomic field. Fermat's primes. Gauss's criterion of constructibility of a regular n-gon.
15 (29/10)	Example: the 17th cyclotomic field. A finite subgroup of the multiplicative group of any field is cyclic.
16 (03/11)	Solvable groups. Examples. The group A₅ is not solvable. Three equivalent definition of solvable groups. Subgroups and quotient groups of solvable groups are solvable. The converse statement.
17 (03/11)	Radical field extensions. Solvable field extensions. Examples of radical and solvable extensions. A Galois extension of a field with enough roots of 1 has a solvable Galois group if and only if the extension is radical.
18 (05/11)	Passing to the Galois / normal closure and adjoining roots of 1. A characteristic zero field extension is solvable if and only if the Galois group of its Galois closure is solvable.
	Reading week, no classes
19 (17/11)	The generic equation of degree n>4 is not solvable in radicals. The equation x⁵-6x+3 is not solvable in radicals. A nonzero polynomial in several variables over an infinite field has a point where it does does not vanish.
20 (17/11)	The primitive element theorem. The normal basis theorem.
21 (19/11)	Discussion of the third homework.
	No classes on Tuesday November 24. Next class is on Thursday November 26.
22 (26/11)	Discussion of the third homework.
23 (01/12)	The normal basis theorem. Towards the Kronecker theorem on computation of the Galois group.
24 (01/12)	Proof of the Kronecker theorem. Reduction mod p and Galois groups. Example of a polynomial of degree n with the Galois group S_n.
25 (03/12)	Outline of other algorithmic methods. Inverse Galois problem. Realisation of some Galois groups of small orders.
26 (08/12)	More on inverse Galois problem. Realisation of the quaternion group. Abelian groups as Galois groups. Statement of Kronecker-Weber theorem. Solvable groups as Galois groups: statement of Shafarevich theorem.
27 (08/12)	Solution of cubic and quartic equations using Galois theory. Criteria for computing Galois groups of quartics.
28 (10/12)	Proof of the fundamental theorem of algebra using Galois theory. Revision of the module.

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

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