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College of Liberal Arts and Sciences University of Florida

MAS 6331–6332 Algebra Qualifying Exam Syllabus

The textbook is David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd edition.

Assumed background knowledge

  • Basic properties of groups, examples, homomorphisms, normal subgroups, and quotients (roughly Sections 1.1–3.2 of Dummit and Foote).
  • Basic properties of rings, examples, ideals, homomorphisms, and quotients (roughly Sections 7.1–7.4).
  • Linear algebra: the definition of a field, vector spaces, and linear transformations, as in a theoretical linear algebra course (roughly Sections 11.1–11.4).

First semester (MAS 6331)

  • Sections 3.3–3.5: Isomorphism theorems, composition series, An.
  • Sections 4.1–4.6: Group actions, Sylow theorems, simplicity of An.
  • Sections 5.1–5.5: Direct and semidirect products of groups.
  • Section 6.1: p-groups, nilpotent groups, and solvable groups.
  • Sections 7.5–7.6: Fraction rings and the Chinese remainder theorem.
  • Sections 8.1–8.3: Euclidean domains, PIDs, and UFDs.
  • Sections 9.1–9.5: Polynomial rings.

Second semester (MAS 6332)

  • Sections 10.1–10.4: Modules and tensor products.
  • Sections 12.1–12.3: Modules over a PID, rational canonical forms, and Jordan canonical forms.
  • Sections 13.1–13.6: Field theory, including algebraic, separable, and cyclotomic extensions.
  • Sections 14.1–14.6: Galois theory, finite fields, and Galois groups of polynomials.