Graduate Algebra I

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: Linear Algebra, Group Theory

Overview:  An overview of graduate algebra with an emphasis on commutative algebra. The instructor may choose 3 or 4 topics from the following, depending on the background and interest of the students.

Detailed Syllabus:

1. Group Actions: Orbit-stabilizer theorem, Centralizers, Normalizers, Class Equation, Sylow Theorems.
2. Rings: Homomorphisms and ideals, quotient, adjunction, integral domains and fraction fields, maximal ideals, factorization, unique factorization domains, principal ideal domains, Euclidean domains, factoring polynomials.
3. Modules: Submodules, quotient modules, homomorphisms, isomorphism theorems, direct sums, simple and semisimple modules, free modules, finitely generated modules, Schur’s Lemma, Jordan-Holder Theorem, Modules over a matrix algebra.
4. Fields: Algebraic and transcendental elements, degree of field extension, adjunction of roots, finite fields, function fields, transcendental extensions, algebraic closure.
5. Galois Theory: Galois group,  Galois extension, cubic equations, symmetric functions, primitive elements, quartic equations, Kummer extensions, quintic equations.
6. Homological Algebra: Categories, monomorphisms and epimorphisms, projective and injective modules, left/right exact functors, additive functors, the Hom functor, diagram chasing, push-outs and pull-backs, tensor product, natural transformations, adjoint functors, flat modules.

References:

1. Algebra by M Artin, 2^nd edition, Prentice-Hall India, 2011.
2. Abstract Algebra by D S Dummit and R M Foote, 2^nd edition, Wiley, 1999.
3. Algebra by S. Lang, 3^rd edition, Springer, 2005.
4. Algebra by T. W. Hungerford, Springer India, 2005.