13. MTH410 (Fall 2020) (Galois theory)
This is an undergraduate course on Galois theory.
There will be tutorials every Thursday (4-4.55).
No internal assessment only end-sem exam (100%).
Timing
Tuesday 2-3 pm (Recorded).
Wednesday 9-10 am (Recorded)
Thursday 4-5 pm (Live-Tutorial)
Text Book
Galois Theory (Joseph Rotman), second edition.
Dummit and Foote
Week 1
01/09/2020: Introduction to the course.
02/09/2020:Motivation for studying Galois theory. Review
from group theory.
03/09/2020:Polynomial rings: Irreducible polynomials, Euclid's algorithms, gcd and lcm of polynomials.
Week 2
08/09/2020:Fields, Prime subfields, different criteria for irreducibility
criteria like Eisenstein, prime subfields.
09/09/2020:Algebraic elements and algebraic extensions.
10/09/2020: Tutorial (Live)
Week 3
15/09/2020:Simple extensions: primitive element theorem.
16/09/2020:Finite fields, Frobenius elements.
17/09/2020: Tutorial (Live): Syllabus-upto ``Algebraic elements and algebraic extensions". Question paper will be sent by email. You may wish to return the answer sheet if you wish (not compulsary).
Week 4
22/9/20: Frobenius automorphism
23/9/20:Splitting fields
24/9/20:Tutorial
Week 5
29/09/2020:Normal extensions (definition).
30/09/2020: Algebraic closure: "Existence and uniqueness of algebraic closure.
01/10/20: Quiz 2
Week 6
October 6:Separable extensions."Normal+separable=Galois.
October 7:Automorphisms and Galois groups.
October8:Tutorial
Week7
October 13:Revision
October 14: Revision
October 15:Short test,
Week 8
October 20:Computations of the Galois groups, Dedekind's Lemma.
October 21:Solvable groups and radical extensions, Galois group of cyclotomic fields.
October 22: Tutorial
Week 9
Teaching break
Week 10
November 3: Fixed fields and subgroups of Galois groups.
November 4: Fundamental theorem of Galois theory.
Week 11
November 10: Some applications of Fundamental theorem of Galois theory:
November 11 :Galois group of composite fields and intersections.
Week 12:
November 17 :Cyclic extensions,
November 18:Fundamental theorem in the context of cyclotomic extension
Week 13
November 24:Norms and traces, Hilbert's Theorem 90.
November 25: Polynomial being solvable by radicals if and only if the Galois group of the splitting field is solvable.
Week 14.
December 1: Detecting when Galois groups are symmetric and Alternating subgroups, resolvent cubic.
December 2:Transitive subgroups of symmetric groups.
Week 15.
December 8:Ruler and compass constructions
December 9: Constructibility of regular n-gon. Assignment 9 uploaded.
Week 16
December 15:Fundamental theorem of Galois theory
in the context of Cyclotomic extensions.
December 16: Abelian extensions of rationals.
Week 17
December 22: Artin-Schreier extensions
December 23: Revision on "ruler and compass construction".