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Syllabus for Preliminary Exam for 631-632 Algebra I, II
Linear algebra: vector spaces, linear transformations, eigenvectors and
diagonalization, Jordan canonical form, bilinear forms and inner product spaces,
normal operators.
Groups: cosets, quotient groups, isomorphism theorems, group actions, Sylow
theorems, finitely generated abelian groups.
Rings: ideals, quotients, isomorphism theorems, principal ideal domains, unique
factorization domains, Euclidean domains, integers and polynomials.

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**Unformatted text preview: **Modules: isomorphism theorems, finitely generated modules over principal ideal domains. Fields: field extensions, algebraic and transcendental elements and extensions, Galois theory. Suggested textbook: M. Artin, Algebra Chapter 2, 3, 4(sections 1-4, 6), 5(sections 5-7), 6(sections 1, 3-6), 7(sections 1-5), 10(sections 1-7), 11(sections 1-4), 12(sections 1-7), 13(sections 1-3,5,6), 14 (sections 1-5)....

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