The topics are divided into six general areas:

- Group Theory
- Nilpotent groups
- Free groups
- Linear groups

- Category theory
- Categories, subcategories
- Functors, equivalence of categories
- Adjoint functors
- Universal properties, representability

- Galois theory
- Algebraic closure
- Algebraic, normal and separable extensions
- Galois correspondence (finite extensions)
- Solvability of equations
- Cubic and quartic equations; cyclotomic fields

- Field theory
- Algebraic and transcendental extensions
- Transcendence basis of an extension

- Commutative ring theory
- Localization; support of a module
- Spectrum of a commutative ring
- Noetherian and Artinian rings
- Hilbert Nullstellensatz
- Hilbert Basis Theorem
- Integral extensions; integral closure
- Associated primes of a module
- Discrete valuation rings; Dedekind domains
- Projective, injective and flat modules; invertible ideals

- Noncommutative ring theory
- Tensor products
- Tensor, symmetric and exterior algebras
- Primitive rings; density theorem
- Semisimple rings
- Wedderburn’s theorem on finite division rings

**Bibliography:**

In recent years, one of the following has served as the core text for the course.

- David S. Dummit and Richard M. Foote,
*Abstract Algebra*, 3rd edition, Wiley - Thomas W. Hungerford,
*Algebra*, Springer Graduate Texts in Mathematics 73 - Serge Lang,
*Algebra*, Springer Graduate Texts in Mathematics 211

Supplementary material can be drawn from the following books.

- Irving Kaplansky,
*Fields and Rings*,

(supplementary material for Galois theory, particularly cubic and quartic equations and cyclotomic fields; supplementary material for noncommutative rings, particularly Noetherian and Artinian rings, the Hilbert Nullstellensatz and Hilbert Basis Theorem) - Hideyuki Matsumura,
*Commutative Ring Theory*, Cambridge University Press (supplementary material in the area of commutative ring theory, particularly the first two and last three subtopics)

**Additional Resources:**

- Richard Foote’s page of errata for Dummit and Foote
- George Bergman’s Companion to Lang’s Algebra with errata
- GMA page on past PhD Algebra Exams

(posted April 11, 2014)