Qualifying exams in algebra

Algebra exam
The algebra qualifying exam covers several standard topics in abstract algebra.
  1. Group theory: basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan–Hölder theorem
  2. 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
  3. Ring theory: factorization in domains, simplicity of matrix algebras
  4. 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
  5. Language of category theory: objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma
  6. Multilinear algebra: pairings, wedge products, symmetric products, multilinear forms over rings
  7. Basic commutative algebra: local rings and localization, integral extensions, Hilbert Basis Theorem, Noether Normalization, Hilbert’s Nullstellensatz
The following textbooks are recommended:
  • Dummit and Foote, Abstract algebra
  • Lang, Algebra
  • Rotman, Advanced Modern Algebra
  • Hungerford, Algebra
  • Jacobson, Basic Algebra I & II
Sample Exams