1. Groups: Finite groups, including Sylow
theorems, p-groups, direct products and sums, semi-direct products,
permutation groups, simple groups, finite Abelian groups; infinite groups,
including normal and composition series, solvable and nilpotent groups,
Jordan-Holder theorem, free groups.
References: Goldhaber Ehrlich, Ch. I except 14;
Hungerford, Ch. I, II; Rotman, Ch. I-VI, VII (first three sections).
2. Rings and modules: Unique factorization
domains, principal ideal domains, modules over principal ideal domains
(including finitely generated Abelian groups), canonical forms of matrices
(including Jordan form and rational canonical form), free and projective
modules, tensor products, exact sequences, Wedderburn-Artin theorem, Noetherian
rings, Hilbert basis theorem.
References: Goldhaber Ehrlich,
Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except
4,6, IV 1,2,3,5,6, VIII 1,4,6.
3. Fields: Algebraic and transcendental
extensions, separable extensions, Galois theory of finite extensions, finite
fields, cyclotomic fields, solvability by radicals.
References: Goldhaber Ehrlich,
Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.
References:
Goldhaber Ehrlich,
Algebra, reprint with corrections, Krieger, 1980.
Hungerford, Algebra,
reprint with corrections, Springer, 1989.
Isaacs, Algebra, a Graduate
Course, Wadsworth, 1994.
Kaplansky, Fields and
Rings, 2nd Edition, University of Chicago Press, 1972.
Rotman, An Introduction
to the Theory of Groups, 4th Edition, W.C. Brown, 1995.