MATH 615 Abstract Algebra
Group Theory
Basic arithmetic properties of Z. Equivalence relations; definitions: group, subgroups, order; homomorphisms, isomorphism; Cosets; Lagrange’s Theorem and applications; cyclic groups; group actions; orbits; stabilizers; orbit partition; normalizers; centralizers; conjugacy; symmetric groups; Cayley’s Theorem; Orbit-Stabiliser Theorem; Sylow’s Theorems; quotient groups; the canonical homomorphism; direct products; finite abelian groups; finite abelian p-groups; the Hall polynomial; Structure Theorem of finitely generated abelian groups.
Ring Theory
Basic examples and definitions; subrings; homomorphisms kernels and Ideals. Quotient Rings. Factorization; Reducibility; roots; special classes of rings; prime and maximal ideals; field extensions; minimal polynomials of finite algebraic extensions; field automorphisms and the Galois Group; the Galois correspondence; fundamental theorem of Galois Theory; a Burly example; application: solution by radicals; insolubility of the quintic.
Prerequisite
Undergraduate courses in fundamentals of mathematical reasoning