Winter 2022

Winter 2022: This semester I am teaching an Algebra course for TIFR-CAM students. It will be an online course taught over Zoom. Meeting ID and Passwords will be sent via emails. An outline of the course is given below.

Title: Algebra.

Lecture Schedule: Monday and Wednesday, 2:30 PM-4:00 PM. The first lecture will be held on January 17th (Monday).

Main Text: We will mainly follow the book `Abstract Algebra (Second Edition)' by Dummit and Foote.

Course Content: The overall plan for the course is the following: (I) Group theory up to Sylow's theorem, Solvable groups, Fundamental Theorem of Finitely Generated Abelian Groups (statement and applications to classification), and if time permits then Semi-direct product. (II) In the Ring Theory we will cover, ideals, prime and maximal ideals, quotient rings, Euclidean domain, Principal Ideal Domain (PID), Unique Factorization Domain (UFD), Polynomial rings, and irreducibility. (III) In the Field theory we will cover field extensions and basic of Galois theory.

More specifically, we will roughly cover the following chapters of the book mentioned above: Chapter 1 (Dihedral group, Symmetric group, are Matrix group, and Quaternion group). We will rigorously do Chapter 4 materials: Group action, Sylow's Theorem, and Simplicity of A_n, these are the core materials from group theory. Bit of Chapter 5 and Solvable group from Chapter 6.

The core topics of ring theory will be Chapter 8 and parts of Chapter 9. In Field theory, we will cover Chapter 13 and parts of Chapter 14, up to the Fundamental Theorem of Galois Theory (with plenty of examples and exercises computing Galois groups). If time permits, then as an application of Galois Theory we prove the unsolvability of degree at least 5 polynomials equations via radicals.

Prerequisites: Students will be expected to know the definition and basic properties of Groups, Subgroups, Cyclic groups, Cosets, Lagrange's Theorem, Normal subgroups, and the First Isomorphism theorem in group theory. In the Ring theory and Field theory, definitions and basic properties and examples of ring, fields, subring, subfield, etc. will be assumed.

Lecture Notes:

Homework: