Course webpage for MTH410: Galois theory (Fall 2015)

Important dates

Quiz 1 August 21, 2015
Quiz 2 September 11, 2015
Mid Sem September 29, 2015
Quiz 3 October 16, 2015
Quiz 4 November 6, 2015
Quiz 5 November 18, 2015
End Sem November 20, 2015


Joseph Rotman : Galois Theory, Second edition
Ian Stewart : Galois Theory
Patrick Morandi : Field and Galois Theory
Serge Lang : Algebra
Dummit and Foote : Abstract Algebra

Distribution of points

Mid Sem 30
End Sem 30
Quizzes: best 4 of 5 quizzes, 10 pts each 40
Total 100


(t) after the date stands for tutorial
Week Date Content
1 August 3 Review of ring theory; introduction to Galois theory
August 5 Fields, field extensions, Examples
August 7 (t) Existence of finite fields of different sizes. Examples of irreducible polynomials. Some properties of prime and maximal ideals. Finite integral domain is a field.
2 August 10 A brief review of ring theory: Definitions, domains, fields, homomorphisms, ideals, quotients. Number of elements in a finite field is a prime power.
August 12 Isomorphism theorems. Polynomial rings over a field. gcd, lcm and their properties.
August 14 (t) Computed kernels of some morphisms. The problem set is here.
3 August 17 Existence of a field extension in which a polynomial splits, prime fields, examples.
August 19 Clarified some doubts; existence of a finite field of size \(p^n\), some lemma's which will lead to Gauss' lemma.
August 21 (t) Quiz 1. Problem set.
4 August 24 Eisenstein criterion. Solution of cubic and quartic polynomials.
August 26 Extensions, splitting fields, adjoining roots. Degree formula
August 28 (t) Problem set.
5 August 31 Lifting isomorphisms to splitting fields.
September 2 Separability. Number of extensions. Galois group.
September 4(t) Problem set.
6 September 7 Galois groups of intermediate extensions. Finite subgroups of the multiplicative group are cycle. Irreduciblility in \(\mathbb{Q}[X]\) and irreducibility mod \(p\)
September 9 Computed \(\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\) and Galois group of cyclotomic extenstions.
September 11(t) Problem set.
7 September 14 Solvable groups and radical extensions
September 16 Polynomial being solvable by radicals implies that the Galois group of the splitting field is solvable.
September 18(t) Problem set. Quiz 2
8 September 28 Mid Sem
September 30 Class postponed because of math symposium.
October 2(t) Holiday: Gandhi Jayanti
9 October 5 Linear independence of characters
October 7 Criteria for an extension to be Galois.
October 9(t) An extra class on some lemmas leading to fundamental theorem of Galois theory, and a tutorial on symmetric functions. Problem set.
10 October 12 Fundamental theorem of Galois theory. Simple extensions and finiteness of number of intermediate fields.
October 14 Some corollaries to the Fundamental theorem. Fundamental theorem of algebra.
October 16(t) Problem set. Quiz 3.
11 October 19 Norms. Norms as determinant of the multiplication map.
October 21 Hilbert's theorem 90 and some corollaries.
October 23 Festival break.
12 October 26 A polynomial is solvable using radicals iff the Galois group is solvable.
October 28 Discriminants and computing Galois groups of splitting fields of low degree polynomials over \(\mathbb{Q}\).
October 31 (t) A set of exercises leading to computing the Galois group of the Artin Schreier polynomial.
13 November 2 Finding Galois group of quartics over \(\mathbb{Q}\). Resolvent cubic.
November 4 Normality and separability in infinite extensions. Infinite Galois extensions.
November 6 (t) Quiz 4.
14 November 9 Some comments on covering space theory, Galois theory and theory of Riemann surfaces.
November 11 Holiday: Diwali
November 13 Ruler and compass constructions: impossibility of trisection of an angle, doubling a cube and squaring a circle.
15 November 16 (t) Some problems regarding computing Galois groups.
November 18 Quiz 5.
November 20 End Sem.