Algebra I

TTR 3:05-4:25, Skiles 271

OFFICE HOURS: Wed 10-11 or by appointment.

Textbook: References:

Topics covered:

Week 1: Groups, subgroups, homomorphisms, normal subgroups, quotient groups, direct products, isomorphism theorems.
Week 2: Normal towers, Abelian and cyclic towers, Jordan-Holder theorem, solvable groups, extensions, semi-direct product. Some examples.
Week 3: Equivalence of internal and external definitions of (semi)-direct products. Lower central series and properties of nilpotent groups. Groups operating on sets. Basic definitions and examples.
Week 4: The orbit and class formulas. p-groups. p-groups are nilpotent. Sylow theory of finite groups. Properties of p-Sylow subgroups. Example of T_1(n,F_p) as a p-Sylow subgroup of Gl(n,F_p).
Week 5: Classification of groups of order p, p^2, pq, p^3 (p,q primes) using semi-direct products. Classification of groups of order 12 and 8. Automorphism groups of cyclic groups. Aut(Z_m) is cyclic when m is a power of an odd prime.
Week 6: Abelian groups. Free Abelian groups. Categories and morphisms.
Week 7: Categorical products and co-products. Universal definitions. Existence of products and co-products in the categories of groups, abelian groups etc. Free groups. Groups defined by generators and relations.
Weeks 8 & 9: Rings. Ideals and quotients. Integral domains. Localization. Local rings. Factorial rings and principal ideal domains. Chinese Remainder Theorem. Idempotents.
Weeks 10 & 11: Modules over commutative rings. Submodules, quotient modules and isomorphism theorems. Free modules and torsion modules. Modules over a PID. Structure theorems for modules over a PID. Modules over an endomorphism of a finite dimensional vectors space as an example of modules over a PID. Rational canonical form and Jordan canonical form. Similarity invariants of a vector space endomorphism, its minimal and characteristic polynomials. The elementary divisor theorem.
Weeks 12 & 13: Fields. Algebraic and transcendental extensions. Existence of algebraic closure. Normal extensions. Splitting fields. Basic properties of finite fields.

Assignments:

Assignment 1 (due Tues, Sept 9). Solutions.
Assignment 2 (due Tues, Sept 23). Solutions.
Assignment 3 (due Tues, Oct 7). Solutions.
Assignment 4 (due Tues, Oct 21). Solutions.
Assignment 5 (due Thus, Nov 13). Solutions.
Assignment 6 (due Tues, Nov 25). Solutions.
Assignment 7 (due Thus, Dec 4).

Final Exam:

Thursday, Dec 11, 11.30 - 2.20pm, Skiles 271.
Saugata Basu