Instructor: Dr Anupam Singh

Audience: BS-MS, MSc students at IISER Pune

Schedule: Lecture: Tuesday 10 AM, Wednesday 10 AM

Tutorial: Friday 11 AM

Evaluation: Test 1 (20%), Mid-Sem (30%), Test 2 (20%), End-Sem (30%)

Prerequisites: Basic knowledge of Group Theory and Linear Algebra.

Goal of the course: Galois theory is one of the crowning glories of Mathematics that paved the way to algebra. The subject originated from the question of solving polynomial equations over a field. Abel and Galois were the main contributors to the subject in the beginning. The concept of "groups" originated from the work of Galois, and it is absolutely essential to understand Algebraic Groups, Algebraic Number Theory, Commutative Algebra, and most of the algebra in modern times. 

Proposed course content:

Field Extensions: Finite, algebraic, and transcendental extensions, adjunction of roots, degree of a finite extension. 

Algebraically closed fields, existence and uniqueness of algebraic closure, splitting fields, normal extensions, separable extensions, Galois extensions.

Automorphism groups and fixed fields, fundamental theorem of Galois theory.

Examples: finite fields, cyclic extensions, cyclotomic extensions, solvability by radicals, ruler and compass constructions, constructibility of regular n-gon.

Text Books :

• Abstract Algebra: Dummit and Foote

• Galois Theory (lectures delivered at the University of Notre Dame): Emil Artin

• Field Theory: Roman

• Galois’ Theory of Algebraic Equations: J.-P. Tignol

• Field and Galois Theory: Morandi

• Galois theory through exercises: Brzezinski

Weekly Schedule:

01/08/2025 Motivation, Three Greek problems, Solving equations by radicals, rings, and fields.

05/08/2025   Ring homomorphism, ideals, maximal and prime ideals, quotient ring

06/08/2025   Polynomial ring, Irreducibility, Gauss' Lemma, Eisenstein Criteria

08/08/2025  Tutorial

    Assignment 1

12/08/2025  Field extension, degree of a field extension, construction of a field extension using irreducible poly

13/08/2025 Multiplicativity of degree

15/08/2025  Holiday

19/08/2025  Algebraic extension, minimal polynomial

20/08/2025  Finite extension vs algebraic extension

22/08/2025  Tutorial

Assignment 2

26/08/2025 Algebraic over algebraic, composite field extension

27/08/2025  Holiday 

29/08/2025  Tutorial 

30/08/2025  Extra Tutorial class

02/09/2025 Straight-edge and Compass construction, constructible number, and impossibility of the three Greek problems

03/09/2025  Splitting field and normal extension

03/09/2025  Test I at 5 PM

05/09/2025  Holiday

09/09/2025  Examples of splitting field

10/09/2025  Algebraic closure, existence of an algebraically closed field

12/09/2025  Tutorial

Assignment 3

16/09/2025  Separable polynomial, Derivative and Separability criteria, Existence of finite fields. 

17/09/2025  Inseparable polynomials, perfect fields, Galois extension. 

19/09/2025  Tutorial

Mid sem exam for the course 22/09/25 at 10 AM

Mid sem exam duration 22-29 September -- no classes

Mid sem break till 1-5th October  -- no classes

07/10/2025  Aut(K/F), subfield-subgroup correspondence, examples

08/10/2025  |Aut(K/F)| bound by degree of extension, Finiteness of Galois group

10/10/2025  Tutorial

Assignment 4

14/10/2025  Linear Independence of Characters

15/10/2025  [K:K^G] = |G|, Galois extension vs Galois group

17/10/2025  Computing Galois Groups

21/10/2025  

22/10/2025

24/10/2025

28/10/2025

29/10/2025

31/10/2025

04/11/2025

05/11/2025  holiday

07/11/2025

11/11/2025

12/11/2025

14/11/2025

End Sem exam on 29/11/2025 at 3 PM