Syllabus for the Preliminary Examination in Algebra
It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.
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.
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.