The algebra qualifying exam covers several standard topics in abstract algebra.

Material

Group theory: basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan–Hölder theorem

Field and Galois theory: finite fields, separable and normal extensions, Fundamental theorem of Galois theory, applications (e.g. solvability by radicals, constructions by straightedge and compass, …), determining Galois groups

Ring theory: factorization in domains, simplicity of matrix algebras

Module theory: basics, projectivity, injectivity, tensor products, flatness, Noetherian property, exact sequences, commutative diagrams, structure theory of modules over a PID, consequences for canonical forms of matrices and other linear algebra

Language of category theory: objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma

Multilinear algebra: pairings, wedge products, symmetric products, multilinear forms over rings

Basic commutative algebra: local rings and localization, integral extensions, Hilbert Basis Theorem, Noether Normalization, Hilbert’s Nullstellensatz