Syllabus

 

Course contents of MAL404: Modern Algebra

Groups: Basic notion of groups - subgroups - cosets of a subgroup - Lagrange's theorem- cyclic groups - permutation groups - normal subgroups - quotient groups - group homomorphisms, isomorphisms and automorphisms - group actions - Cayley’s theorem - Sylow's theorem - direct products of groups- finite abelian groups; Rings: definition and examples of rings - subrings - Ideals - maximal and prime ideals - quotient rings- ring homomorphisms and isomorphisms. Integral domains: division rings and fields - field of quotients of an integral domain - Euclidean domains - principal ideal domains, unique factorization domains - Polynomial ring- Irreducibility of polynomials; Fields: Subfields - extension fields - algebraic extensions- roots of a polynomial - splitting fields- algebraically closed field - normal and separable extensions - Ruler and compass constructions. Galois theory: Fundamental theorem of Galois theory - polynomials solvable by radicals.