MATH-660 Graduate Algebra I

Group theory Sylow theorems. Solvable and nilpotent groups, normal and central series, free groups, simple groups, Jordan-H¨older theorem. Direct and semi-direct products, extensions. Category theory Categories and functors, natural transformations, universal properties, products and coproducts. Rings and modules ◦ Polynomial rings, elementary symmetric polynomials. Euclidean, Principal ideal, and Unique factorization domains, Gauss lemma. ◦ Structure theorem for modules over PID: elementary divisors and invariant factor forms. Noetherian and Artinian rings and modules, Hilbert basis theorem, simple modules, composition series, and Jordan-H¨older theorem for modules. ◦ Vector spaces and linear operators, characteristic and minimal polynomials, Cayley-Hamilton theorem, canonical Jordan form, Rational Canonical Form. ◦ Semi-simple rings, Artin-Wedderburn theorem. Prerequisites (Undergraduate courses in Abstract Algebra)

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