MTH 410 (Galois theory)

This is an undergraduate course on Galois theory. Please read the course template (attachment) for text-books and other important information about the course.

There will be tutorials every Thursday (11-11.55). There will be short tests on the tutorials of 30 minutes duration starting from 11/08/2016.

Since continuous assessment is based on 10 tests, your short test marks will be updated every week in this website as a file MTH410.pdf.

If you see any anomaly in the file and the actual marks that you got

in the short test, kindly contact me immediately to get it rectified.

This file will be removed after the last short test.

Office hours

Thursday 3-4 pm.

Text Book

Galois Theory (Joseph Rotman), second edition.

Grading :

1. Mid-Semester Exams-30%
2. End-Semester Exams-30%.
3. Short tests-24% (best of 6 tests)
4. Class tests (2):-8%+8%.

Please note that there will not be any repeat class tests. If you are absent in any of the two class tests then we will use the short test points that are NOT used in the short test performances (1 class test=2 short tests).

First class test will be on 01/09/2016.

Second class test will be on 03/11/2016.

Week 1

01/08/2016: Motivation for studying Galois theory. Review

from group theory (symmetric groups and solvability), ring theory.

02/08/2016: Polynomial rings: Irreducible polynomials, Euclid's algorithms, gcd and lcm of polynomials.

04/08/2016: Fields, Prime subfields, different criteria for irreducibility

like Eisenstein, prime subfields.

04/08/2016: Assignment 1 uploaded.

Week 2

08/08/2016: Algebraic elements and algebraic extensions.

09/08/2016: Simple extensions: primitive element theorem,

Simple algebraic and transcendental extensions, Luroth's Theorem (statement).

11/08/2016: Tutorial, short test.

12/08/2016:Assignment 2 (uploaded).

Week 3

15/08/2016: Independence day.

16/08/2016: Finite fields, Frobenius elements

18/08/2016: Tutorial, short test.

Week 4

22/08/2016: Tutorial.

23/08/2016:Splitting fields and normal extensions (definition).

24/08/2016: Tutorial, short test. MTH410.pdf uploaded.

25/08/2016: Assignment 4 uploaded.

Week 5

29/08/2016: Separable extensions.

"Normal+separable=Galois."

30/08/2016:Existence and uniqueness of algebraic closure.

01/09/2016: Class test.

Week 5

05/09/2016:Ganesh Chaturthi.

06/09/2016: Automorphisms and Galois groups.

07/09/2016: Assignment 5 uploaded.

08/09/2016:Short test, tutorial. MTH410.pdf uploaded.

Week 6

12/09/2016:Idu’l Zuha (Bakrid)

13/09/2016:Revision, computations of the Galois groups, Dedekind's Lemma.

15/09/2016:Short test, tutorial.

Week 7

Mid-sem exam.

Week 8

26/09/2016:Tutorial on the mid-sem exam,

27/09/2016:Solvable groups and radical extensions, Galois group of cyclotomic fields.

29/09/2016: Fixed fields and subgroups of Galois groups.

30/09/2016:Assignment 6 and MTH410.pdf uploaded.

Week 9

03/10/2016:Fundamental theorem of Galois theory.

04/10/2016: Some applications of Fundamental theorem of Galois theory:

1. Galois group of composite fields and intersections,

2. The existence and uniqueness of Galois closure

06/10/2016:Short-test, tutorial. MTH410.pdf uploaded.

Week 10

Festival break.

13/10/2016:Assignment 7 uploaded.

Week 11

17/10/2016:Cyclic extensions, Norms and traces, Hilbert's Theorem 90.

18/10/2016: Polynomial being solvable by radicals if and only if the Galois group of the splitting field is solvable.

20/10/2016: Short-test, tutorial.

Week 12.

24/10/2016: Detecting when Galois groups are symmetric and Alternating subgroups, resolvent cubic.

25/10/2016:Transitive subgroups of symmetric groups.

Assignment 8 uploaded.

Announcement: Please have a look at the following webpage

with very nice material on Galois theory:

1. Professor Keith Conrad

http://www.math.uconn.edu/~kconrad/blurbs/

2. Dr. Anupam Singh

https://sites.google.com/site/anupamk182/galoistheory%28iiser%2Cpune%29

27/10/2016:Short-test, tutorial. MTH410.pdf uploaded.

Week 13.

31/10/2016:Ruler and compass constructions

1/11/2016: Constructibility of regular n-gon. Assignment 9 uploaded.

03/11/2016:Class test.

Week 14.

07/11/2016:Fundamental theorem of Galois theory

in the context of Cyclotomic extensions.

Please have a look at the wikipedia page:

https://en.wikipedia.org/wiki/Cyclotomic_polynomial

for some general knowledge

08/11/2016: Abelian extensions of rationals, Artin-Schreier extensions, revision.

Assignment 10 uploaded.

10/11/2016: Short test, tutorial.

Announcements:

1. MTH410.pdf uploaded.

2. If you get a cubic polynomial first check that it is irreducible

or not then try to do anything else.

3. Please attend the class of 17/11/2016 where your final internal marks will be entered in the marksheet in front of you and there

will be a photo session.

Week 15.

14/11/2016:Guru Nanak's birthday (Holiday).

15/11/2016: Infinite Galois extensions: Please note that

the last exam will be based on "ruler and compass construction".

16/11/2016: There was a mistake in the assignment. Thank you so much Tanushree and Sudhir for pointing it out to me.

Please find an updated assignment10.

17/11/2016:Short test, photo session, final internal marks entry.

Thank you very much for taking the course. Hope you had enjoyed the acheivement of Galois. Please find our

photos and the marks (without end-sem).

Week 16.

End semester exam.