Simple groups and solvable groups, nilpotent groups, simpcity of alternatinlig groups, composition series, JordanHolder Theorem.Semidirect products. Free groups, abelian groups.
Rings,Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions,Chinese Remainder Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Polynomial rings over UFD's.
Fields, Characteristic and prime subfields, Field extensions, Finite,algebraic and finitely generated field extensions, Classical ruler and compass constructions, Splitting fields and normal extensions, algebraic closures. Finite fields, Cyclotomic fields, Separable and inseparable extensions.
Galois groups, Fundamental Theorem of Galois Theory, Composite extensions, Examples (including cyclotomic extensions and extensions of finite fields). Solvability by radicals, Galois' Theorem on solvability. Cyclic and abelian extensions, transcendental extensions.