| Week |
Date |
Content |
| 1 |
August 3 |
Review of ring theory; introduction to Galois theory |
| August 5 |
Fields, field extensions, Examples |
| August 7 (t) |
Existence of finite fields of different sizes. Examples of
irreducible polynomials. Some properties of prime and maximal
ideals. Finite integral domain is a field. |
| 2 |
August 10 |
A brief review of ring theory: Definitions, domains, fields,
homomorphisms, ideals, quotients. Number of elements in a finite
field is a prime power.
|
| August 12 |
Isomorphism theorems. Polynomial rings over a field. gcd, lcm and
their properties.
|
| August 14 (t) |
Computed kernels of some morphisms. The problem set is here. |
| 3 |
August 17 |
Existence of a field extension in which a polynomial splits, prime
fields, examples. |
| August 19 |
Clarified some doubts; existence of a finite field of size
\(p^n\), some lemma's which will lead to Gauss' lemma. |
| August 21 (t) |
Quiz 1. Problem set.
|
| 4 |
August 24 |
Eisenstein criterion. Solution of cubic and quartic polynomials. |
| August 26 |
Extensions, splitting fields, adjoining roots. Degree formula |
| August 28 (t) |
Problem set. |
| 5 |
August 31 |
Lifting isomorphisms to splitting fields. |
| September 2 |
Separability. Number of extensions. Galois group. |
| September 4(t) |
Problem set. |
| 6 |
September 7 |
Galois groups of intermediate extensions. Finite subgroups of the
multiplicative group are cycle. Irreduciblility in \(\mathbb{Q}[X]\)
and irreducibility mod \(p\) |
| September 9 |
Computed \(\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\) and
Galois group of cyclotomic extenstions. |
| September 11(t) |
Problem set. |
| 7 |
September 14 |
Solvable groups and radical extensions |
| September 16 |
Polynomial being solvable by radicals implies that the Galois
group of the splitting field is solvable. |
| September 18(t) |
Problem set. Quiz 2 |
| 8 |
September 28 |
Mid Sem |
| September 30 |
Class postponed because of math symposium. |
| October 2(t) |
Holiday: Gandhi Jayanti |
| 9 |
October 5 |
Linear independence of characters |
| October 7 |
Criteria for an extension to be Galois. |
| October 9(t) |
An extra class on some lemmas leading to fundamental theorem of
Galois theory, and a tutorial on symmetric functions. Problem set. |
| 10 |
October 12 |
Fundamental theorem of Galois theory. Simple extensions and
finiteness of number of intermediate fields. |
| October 14 |
Some corollaries to the Fundamental theorem. Fundamental theorem
of algebra. |
| October 16(t) |
Problem set. Quiz 3. |
| 11 |
October 19 |
Norms. Norms as determinant of the multiplication map. |
| October 21 |
Hilbert's theorem 90 and some corollaries. |
| October 23 |
Festival break. |
| 12 |
October 26 |
A polynomial is solvable using radicals iff the Galois group is
solvable. |
| October 28 |
Discriminants and computing Galois groups of splitting fields of
low degree polynomials over \(\mathbb{Q}\). |
| October 31 (t) |
A set of exercises leading to computing the Galois group of the
Artin Schreier polynomial. |
| 13 |
November 2 |
Finding Galois group of quartics over \(\mathbb{Q}\). Resolvent
cubic. |
| November 4 |
Normality and separability in infinite extensions. Infinite Galois
extensions. |
| November 6 (t) |
Quiz 4. |
| 14 |
November 9 |
Some comments on covering space theory, Galois theory and theory
of Riemann surfaces. |
| November 11 |
Holiday: Diwali |
| November 13 |
Ruler and compass constructions: impossibility of trisection of an
angle, doubling a cube and squaring a circle. |
| 15 |
November 16 (t) |
Some problems regarding computing Galois groups. |
| November 18 |
Quiz 5. |
| November 20 |
End Sem. |