Algebra
TIFR-CAM, January-April, 2023.

Classes will be held from 11:00 AM to 12:20 PM on Wednesdays and Fridays starting from 4th of January. The classes will be held partially over zoom and partially in person.
Mondays (11:00 AM-12:30 PM) will be left to casual discussions and recitations. If you have questions, please ask your queries by Friday so that they can be taken up on Monday.
Grading Scheme: 40 (Final term)+30 (midterm)+ 30 (assignments and recitation); solutions from bonus problems and recitations can be used to improve the net midterm+assignment score.
Here is the conversion chart from marks to letter grades. More about the grading scheme can be found by following this link.
Letter Grade Marks range
O 95-100
A+ 90-94
A 85-89
B+ 75-84
B 60-74
C 50-59
P 40-49
F Below 40
We might take inspiration from the books by Artin "Algebra" and notes by Lior Silberman "Link" .
The midterm will be held on March 6, 2023 from 11:00-12:30. There maybe some student lectures (for extra credit) in case they are enthusiastic.
Lecture notes (videos if any will be available only for 2-3 days. Please download them before I delete them. Further the notes are at best rough guidelines.)

Lecture 1, date: 4 January, 2023 Notes Introduction and random banter

Lecture 2, date: 6 January, 2023 Notes , Groups as symmetries

Lecture 3, date: 11 January, 2023 Notes symmetric groups, semidirect producs

Lecture 4, date: 13 January, 2023 Notes Groups actions, free groups

Lecture 5, date: 18 January, 2023 Notes Presentations of a group and a description of the normal closure

Lecture 6, date: 20 January, 2023 Notes Continuing with group presentations

Office hour on 23 January, 2023; Clarified the universal property and spoke about the undecidability of the word problem

Lecture 7, date: 25 January, 2023 Notes Cayley graphs

Lecture 8, date: 27 January, 2023 Notes Sabidussi's theorem

Lecture 9, date: 1 February, 2023 Notes Schreier's theorem

Lecture 10, date: 3 February, 2023 Notes Cauchy's theorem

Lectures 11 and 12, date: 8 and 10 February, 2023 Notes Sylow theorem

Lecture 13, date: 22 February, 2023 Notes Groups of order 21, finite generated abelian groups and rings Video Passcode: Uc#75jBZ

Lecture 14, date: 24 February, 2023 Notes Rings and modules, structure theorem for finitely generated abelian groups Video Passcode: r48P&E

Lecture 15 and 16, date: 1 and 3 March, 2023 Notes structure theorem for finitely generated abelian groups

Lecture 15 and 16, date: 1 and 3 March, 2023 Notes Structure theorem for finitely generated abelian groups

Lectures 17 to 22, date: 8 to 24 March, 2023 Notes Decomposition Series and Solvable Groups

Lectures 23 to 24, date: 29 to 31 March, 2023 Notes Polynomials, factorisation and other such criterion

Lectures 25, date: 6 April, 2023 Notes Maximal and prime ideals

Lectures 26 to 29, date: 12 April to 21 April, 2023 Notes Irreducible polynomials, constructible numbers and splitting fields

Lectures 30, date: 26 April, 2023 Notes Normal extensions

Lectures 31, date: 28 April, 2023 Notes Main theorem of Galois

Assignments

Please feel free to discuss with your fellow students but the solution must be understood and written on your own. Please do not search for the solutions on the internet. It helps no one (including you). The assignment needs to submitted by email to my Tifr address before the class (scan/tex either will do) or submitted to me in class.

Assignment 1, due on 1st February, 2023: Link

Assignment 2, due on midnight 23rd February, 2023: Link

Assignment 3, due on 15 March, 2023: Link

Assignment 4, due on 12 April, 2023: Link

Assignment 5, due on 21 April, 2023: Link

Assignment 6, due on 28 April, 2023: Link

Some practice problems for material not covered by assignments.

Some Resources

Galois and group theory by G. Birkhoff Link

Every set can be given a group structure assuming the Axiom of Choice Link

A classification of Dedekind/Hamiltonian groups Link The book by Marshall Hall on the theory of groups provides a proof

(Non)-Solvability of the symmetric group of 5 elements and Barrington's theorem Link

Subgroups of p-subgroups (from discussions in class and typed by Aadi Bhure) Link

Finite fields and linear codes (by Suayb Arslan) Link