MT3174
This is an undergraduate course on Galois theory.
Venue-LHC 206
Assessment:
End-sem-50%
Continuos evaluation:50% (10%+10%+10%+10%+10%)
Timing
Wednesday -4-5 pm
Thursday 3-4 pm
Friday 11-12.
Text Book
Dummit and Foote
Galois Theory (Joseph Rotman), second edition.
Week 1
10/08/2022:Motivation for studying Galois theory. Review
from group theory.
11/08/2022:Polynomial rings: Irreducible polynomials, Euclid's algorithms, gcd and lcm of polynomials.
12/08/22:Fields, Prime subfields, different criteria for irreducibility
criteria like Eisenstein, prime subfields.
Week 2
17/08/2022:The instructor spoke in https://icts.res.in/program/ecl2022
Mr. Ronit Debnath took tutorial.
18/08/22:Algebraic elements and algebraic extensions.
19/08/22:Simple extensions: primitive element theorem.
Week 3
24/08/22:Finite fields,
25/08/22:Frobenius elements, Frobenius automorphism and Splitting fields
26/08/22:Tutorial
Week 4
31/08/22: Holiday
01/09/22:Normal extensions (definition).
02/09/22:Quiz 1 (Syllabus upto Splitting fields)
Week 5
07/09:Algebraic closure: "Existence and uniqueness of algebraic closure.
08/09:Separable extensions."Normal+separable=Galois.
Tutorial
Week 6
14/09/22:Computations of the Galois groups, Dedekind's Lemma.
15/09/22:Automorphisms and Galois groups.
16/09/22:Tutorial
Week7
21/09/22:Quiz 2 (syllabus-uptoGalois extensions).
22/09/22:Solvable groups and radical extensions
23/09/22: No class, study leave for mid-semester exam.
Week 8
Teaching break
Week 9
06/10: Fixed fields and subgroups of Galois groups
07/10: Galois group of cyclotomic fields.
Week 10
12/10: Quiz 3 (Syllabus:Material taught before teaching break)
13/10:Fundamental theorem of Galois theory.
14/10:Some applications of Fundamental theorem of Galois theory.
Week 11
19/10:Cyclic extensions,Norms and traces, Hilbert's Theorem 90.
20/10: Polynomial being solvable by radicals if and only if the Galois group of the splitting field is solvable.
21/10:Tutorial
Week 12:
26/10:Quiz 4.
27/10: Tutorial (by Mr. Pranjal Vishwakarma)
28/10: Detecting when Galois groups are symmetric and Alternating subgroups, resolvent cubic.
Week 13
02/11: Transitive subgroups of symmetric groups.
03/11: Artin- Schreier extension.
04/11: TUTORIAL
Week 14.
09/11:Ruler and compass constructions
10/11: Quiz 5
11/09: Constructibility of regular n-gon.
Week 15.
16/09: Revision
17/09:Fundamental theorem of Galois theory
in the context of Cyclotomic extensions.
18/09: Infinite Galois extension (no question will be asked in end-sem).
Week 16
Study leave
Week 17
End sem exam