Graduate Studies Committee Proposal for
the Algebra Comprehensive

Existing guideline:

Groups: symmetric and alternating groups, Sylow theorems, solvable groups, structure of finite abelian groups.

Linear Algebra: eigenvalues and eigenvectors, canonical forms for matrices, determinants, Gaussian elimination, Cayley-Hamilton theorem, diagonalization of real symmetric matrices. 

Commutative Rings: Euclidean rings, rings of fractions, principal ideal and unique factorization domains, polynomial rings, Gauss' lemma, Eisenstein's criterion, rings of algebraic integers.

Fields: splitting fields, Galois theory, separable polynomials, finite fields, solvability by radicals. Finite-dimensional algebras: radical of a ring, simple algebras, Wedderburn's theorems, group algebras, quaternion algebras.

Basic Courses: 525, 526, 581, 582.

References:

1. Introduction to Abstract Algebra by E. A. Walker
2. Topics in Algebra by I. Herstein
3. Basic Algebra I and II by N. Jacobson
4. Algebra by T. Hungerford.

Proposed guideline:

Groups: symmetric and alternating groups, Sylow theorems, structure of finite abelian groups.

Linear Algebra: eigenvalues and eigenvectors, canonical forms for matrices, determinants, Gaussian elimination, Cayley-Hamilton theorem.

Commutative Rings: rings of fractions, principal ideal and unique factorization domains, polynomial rings, Gauss' lemma, Eisenstein's criterion, chain conditions, Dedekind domains.

Modules: free and projective modules, homomorphisms, tensor products, localization.

Fields: splitting fields, Galois theory, separable polynomials, finite fields, solvability by radicals.

Finite-dimensional algebras: radical of a ring, simple algebras, Wedderburn's theorems.

Basic Courses: 525, 526, 581, 582.

References:

1. Introduction to Abstract Algebra by E. A. Walker
2. Topics in Algebra by I. Herstein
3. Algebra by T. Hungerford
4. Field and Galois Theory by P. Morandi
5. Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald.