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Every fall semester
Graduate level linear and abstract algebra including groups, rings, modules, and fields. (1st of two courses)
Text at the level of Abstract Algebra by Dummit and Foote.
Groups: Basic facts about groups, including cyclic, dihedral, symmetric, and linear (matrix) groups; homomorphisms; cosets and quotients; normal subgroups; mapping properties and isomorphism theorems; group actions; applications of group actions to geometric and combinatorial symmetry (e.g. symmetries of regular polyhedra, Polya enumeration, and Burnside's formula); the class equation; the Sylow theorems; simple groups and composition series; the structure theorem for finitely generated abelian groups.
Rings: (By a ring we mean a commutative ring with identity.) Definitions; homomorphisms; ideals; quotients; mapping properties and isomorphism theorems; polynomial rings; integral domains; fraction fields; prime and maximal ideals; Euclidean domains; unique factorization domains; principal ideal domains; Gauss's lemma; irreducibility criteria; the Chinese remainder theorem.
Modules: Definitions and examples; homomorphisms and quotients; generation, freeness, and finiteness properties; the structure theorem for modules over a principal ideal domain.
Fields: Characteristic; prime fields; field extensions; algebraic vs. transcendental extensions; splitting fields; basic properties of finite fields; algebraic closure.
Galois Theory: Definitions; the fundamental theorem of Galois theory; examples, including quadratic, cubic, cyclotomic, and finite fields; the primitive element theorem.