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MTH-205.4 Algebra

Groups, subgroups, homomorphisms, normal subgroups, quotient groups,

isomorphism theorems, symmetric groups, alternating and dihedral groups.

Structure of finitely generated abelian groups, group actions and its applications, Sylow theorems, solvable groups.

Rings and homomorphisms, ideals, isomorphism theorems, prime ideals and maximal ideals, Jacobson radical and nil-radical, Chinese remainder theorem, polynomial rings and power series rings, division algorithm, roots and multiplicities,

Resultant and discriminant, elementary symmetric functions and the main theorem on symmetric functions, proof of the fundamental theorem of algebra by using symmetric functions, factorization in polynomial rings, Eisenstein criterion, unique factorization domains.

Modules, homomorphisms and exact sequences, free modules, rank of a free module (over commutative rings), hom and tensor products, chain conditions on Modules, Noetherian rings and Hilbert basis theorem, structure theorem for modules over PIDs.

Field extensions and elementary Galois theory.