Algebra Oral Qualifying Exam Syllabus

algebra is the study of fundamental algebraic structures that occur throughout mathematics: groups, rings, modules, categories, and fields—as well as maps between them. A solid grasp of algebra can be useful to students of all mathematical stripes, and accordingly the suggested syllabus is a “strong core plus more” model: a student preparing for an oral qualifying exam in algebra should (1) feel confident with the core topics (covered in most offerings of 101/111) and (2) should select a supplementary advanced topic for examination, in consultation with the chosen oral exam committee. The core and some suggested supplementary topics are listed below. 1. Core topics covered in most offerings of 101/111 (1) Groups (e.g., [4, Chapters 1–5], [15, Chapters 1-6]) (a) Cyclic, abelian, non-abelian groups; normal subgroups and quotient groups; normal series, solvable groups, commutators, composition series, Jordan-Hölder theorem; the extension problem (b) Group actions, G-sets and structure theorem, Sylow theory, semi-direct products and split exact sequences (c) Symmetric group, alternating subgroup, cycle decomposition and conjugacy classes (2) Rings (e.g., [4, Chapter 7–9], [10, Chapters 2,4], [15, Chapters 7-8]) (a) Rings: examples, properties, homomorphisms, group rings, polynomial rings, division algorithm, polynomials versus polynomial maps (b) Ideals: prime and maximal ideals, operations on ideals, Chinese remainder theorem (CRT) (c) Irreducibles and prime elements, UFDs, PIDs, Noetherian rings, Euclidean domains, Gauss’s lemma and corollaries, irreducibility tests, Hilbert’s Basis Theorem, cyclotomic polynomials (3) Modules and categories (e.g., [3, Chapters 2–3], [4, Chapters 10–12, Appendix II], [10, Chapter 3]) (a) Vector spaces and linear algebra (b) Modules: simple, indecomposable, free, isomorphism theorems (c) Basic category theory: universal definitions, functors (d) Exact sequences, sections, retractions; and connections to free and projective modules (e) Products and coproducts of modules: Hom(·, ·) as a functor and exactness properties, and Hom( ⊕ i Mi, ∏ j Nj) (f) Localization, tensor products, extension of scalars, flat modules; and connections with coproducts, pullbacks, pushouts (g) Finitely generated modules over a PID: general structure theory, linear operators and vector spaces as k[x]-modules, applications to canonical forms Date: July 10, 2017.