- Groups: definitions & basic examples;
- Normal subgroups, quotients;
- Three isomorphism theorems;
- Centralizer and normalizer of a subset, centre of a group;
- Permutations, symmetrc groups and Cayley's theorem;
- Group actions and their applications, Sylow's theorems.
- Rings and ideals: basic definitions, quotient rings;
- The Chinese Remainder Theorem;
- Maximal and prime ideals;
- Unique factorization, unique factorization domains, principal ideal domains, Euclidean domains, polynomial rings;
- Modules: basic definitions and examples, Hom and tensor products, the Structure Theorem for finitely generated modules over PIDs;
- Fields: basic definitions and examples, algebraic & trancendental numbers;
- Finite fields, characteristic, the order of a finite field.
Artin, M., Algebra, Prentice-Hall of India, 1994.
Dummit, D. S. and Foote, R. M., Abstract Algebra, John Wiley & Sons, 2001.
Herstein, I. N., Topics in Algebra, John Wiley & Sons, 1995.
Lang, S., Algebra, 3rd editiom. Springer, 2002.