Fields and Galois Theory
MT 3174, IISER Pune
August 2021 Semester
Instructor: Dr Anupam Singh
Audience: BS-MS students at IISER Pune
Schedule: Lecture: Wednesday 3 PM, Thursday 12 Noon (on classroom through google meet)
Tutorial: Friday 11 AM (via google meet)
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 main contributors to the subject in 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, Wiley India.
Galois Theory (lectures delivered at the University of Notre Dame): Emil Artin, Dover
Field Theory: Roman, Springer.
Galois’ Theory of Algebraic Equations: J.-P. Tignol, World Scientific.
Weekly Schedule:
18/08/2021 Definition of field and examples, Motivating examples from Greek geometry - straightedge & compass constructions
19/08/2021 Holiday
20/08/2021 Field homomorphism, Characteristics of a field
25/08/2021 Polynomial ring, irreducibility - Gauss' Lemma
26/08/2021 Eisenstein criteria and application
27/08/2021 Tutorial
01/09/2021 Degree of a field extension, construction of an extension given an irreducible polynomial
02/09/2021 Field generated by elements, multiplicativity of degree
03/09/2021 Tutorial
08/09/2021 Algebraic elements, algebraic extensions
09/09/2021 Finite extensions, splitting field
10/09/2021 Holiday
15/09/2021 Relation between finite extension and algebraic extension
16/09/2021 Algebraic over algebraic, composite field
17/09/2021 Tutorial
22/09/2021 Splitting field, Normal Extension, Separable Polynomial
23/09/2021 Criteria for separability, Inseparable polynomials, Separable Extensions, Galois Extension
24/09/2021 Tutorial
29/09/2021 Straightedge and Compass constructions, constructible numbers and impossibility
30/09/2021 Tutorial
01/10/2021 Tutorial
04-14 October 2021 Mid Sem Exam
20/10/2021 Aut(K/k), Galois group of a polynomial
21/10/2021 Relation between K/k and Aut(K/k)
22/10/2021 Tutorial
27/10/2021 Fundamental Theorem of Galois Theory: Statement, Extending field isomorphisms
28/10/2021 |G|=[K:K^G].
29/10/2021 Tutorial
03/11/2021 Fixed field of a subgroup of Galois group
04/11/2021 Holiday
05/11/2021 Tutorial
10/11/2021 Existence of algebraic closure
11/11/2021 Proof of fundamental theorem of Galois theory
12/11/2021 Tutorial
17/11/2021 Radical extensions, Abelian extensions
18/11/2021 Solvable groups, Solving equations by radicals
19/11/2021 Holiday
20/11/2021 Test 2
24/11/2021 Finite fields
25/11/2021 Galois theory of finite fields
26/11/2021 Tutorials
01/12/21 Tutorials
02/12/21 No class/ Discussions
03/12/21 No class / Discussions
7-17 December 2021 End Sem exam