This is an undergraduate course on Galois theory.
Venue-LHC 205
Assessment:
Mid Sem-30%
End-sem-30%
Continuos evaluation:40% (10%+10%+10%+10%)
Timing
Tuesday 10-11 am
Wednesday -10-11 am
Friday 11-12.
Text Book
Dummit and Foote
Galois Theory (Joseph Rotman), second edition.
Week 1
Motivation for studying Galois theory.
Review
from group theory.Polynomial rings: Irreducible polynomials, Euclid's algorithms, gcd and lcm of polynomials.
Fields, Prime subfields, different criteria for irreducibility
criteria like Eisenstein, prime subfields.
Week 2
Algebraic elements and algebraic extensions.
2. Simple extensions: primitive element theorem.
Week 3
Finite fields,
Frobenius elements, Frobenius automorphism and Splitting fields
Week 4
Normal extensions (definition).
Quiz 1 (Syllabus upto Splitting fields)
Week 5
Algebraic closure: "Existence and uniqueness of algebraic closure.
Separable extensions."Normal+separable=Galois.
Week 6
Computations of the Galois groups, Dedekind's Lemma.
Automorphisms and Galois groups.
Week7
Quiz 2 (syllabus-uptoGalois extensions).
Solvable groups and radical extensions
No class, study leave for mid-semester exam.
Week 8
Teaching break
Week 9
Fixed fields and subgroups of Galois groups
Galois group of cyclotomic fields.
Week 10
Quiz 3 (Syllabus:Material taught before teaching break)
Fundamental theorem of Galois theory.
Some applications of Fundamental theorem of Galois theory.
Week 11
Cyclic extensions,Norms and traces, Hilbert's Theorem 90.
Polynomial being solvable by radicals if and only if the Galois group of the splitting field is solvable.
Week 12:
Quiz 4.
Detecting when Galois groups are symmetric and Alternating subgroups, resolvent cubic.
Week 13
Transitive subgroups of symmetric groups.
Artin- Schreier extension.
Week 14.
Ruler and compass constructions
Constructibility of regular n-gon.
Week 15.
Fundamental theorem of Galois theory
in the context of Cyclotomic extensions.
Infinite Galois extension (no question will be asked in end-sem).
Week 16
Study leave
Week 17
End sem exam