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Department:
MATH
Course Number:
6121
Hours  Lecture:
3
Hours  Lab:
0
Hours  Recitation:
0
Hours  Total Credit:
3
Typical Scheduling:
Every fall semester
Graduate level linear and abstract algebra including groups, rings, modules, and fields. (1st of two courses)
Course Text:
Text at the level of Abstract Algebra by Dummit and Foote.
Topic Outline:

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.