Ph.D. Mathematics Preliminary Exam Syllabi

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.

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.