MTL501: Algebra

4 Credits (3-1-0)

Groups, subgroups, Lagrange theorem, quotient groups, isomorphism theorems; cyclic groups, dihedral groups, symmetric groups, alternating groups; simple groups, simplicity of alternating groups; Group action, Sylow theorems and applications; free abelian groups, structure of finitely generated abelian groups; Solvable and nilpotent groups, composition series, Jordan-Holder theorem.

Rings, examples: polynomial rings, formal power series, matrix rings, group rings; prime ideals, maximal ideals, quotient rings, isomorphism theorems; Integral domains, PID, UFD, Euclidean domains, division rings, field of fractions; primes and irreducibles, irreducibility criteria; product of rings, Chinese remainder theorem.

Field extension, algebraic extension, algebraic closure, straight edge and compass constructions, splitting fields, separable and inseparable extensions, fundamental theorem of Galois theory; solvability by radicals.