## Syllabus for the Comprehensive Exam in Algebra

1. Vector Spaces (Linear Algebra): Vector spaces and bases; matrices and linear transformations; eigenvalues and eigenvectors; trace and determinant; minimal and characteristic polynomials; the Cayley-Hamilton theorem; inner products and norms; orthogonality and orthogonal projections; bilinear and quadratic forms; the spectral theorem for symmetric matrices; variational characterization of eigenvalues; diagonalization; rational and Jordan canonical forms over arbitrary fields; signature and Sylvester's law of inertia.
2. 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.

3. 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.

4. Modules: Definitions and examples; homomorphisms and quotients; generation, freeness, and finiteness properties; the structure theorem for modules over a principal ideal domain.

5. Fields: Characteristic; prime fields; field extensions; algebraic vs. transcendental extensions; splitting fields; basic properties of finite fields; algebraic closure.

6. Galois Theory: Definitions; the fundamental theorem of Galois theory; examples, including quadratic, cubic, cyclotomic, and finite fields; the primitive element theorem.

Suggested textbook: Abstract Algebra by Dummit and Foote
Suggested courses: 6112 and 6121
Other relevant courses: 4107, 4108, and 4305