algebra

The Algebra Qualifying Exam will be based on the following syllabus (updated in Fall 2019). The sequence Math 210ABC will prepare students according to this syllabus.

Group Theory: basic notions and results (e.g., homomorphisms, subgroups, quotient groups, cosets, conjugation, generators, Lagrange’s theorem), isomorphism theorems,automorphism groups, symmetric groups, linear groups, group actions and permutation representations, the Sylow theorems and applications to simple groups, free groups and presentations, composition series and the Jordan-Holder theorem, nilpotent and solvable groups, semidirect products and group extensions.

Ring and Module Theory: basic notions and results (e.g., homomorphisms, ideals and generators, subrings, quotient rings), maximal and prime ideals, units, division rings, fields of fractions, PIDs, UFDs, polynomial rings and factorization (e.g., Gauss’s lemma, Eisenstein criterion), structure of finitely generated modules over a PID, canonical forms, duality and bilinear pairings, tensor products, localization, Nakayama’s lemma, torsion and rank, chain conditions and noetherian and artinian rings and modules, exterior powers and determinants.

Field and Galois Theory: basic notions and results, algebraic and transcendental extensions, separable, inseparable and normal extensions, field embeddings, algebraic closure, Galois theory, cyclotomic extensions, finite fields, solvability by radicals, norm and trace, discriminants of polynomials, Hilbert’s Theorem 90, Kummer theory.

Category Theory: basic definitions and examples, full and faithful functors, monomorphisms and epimorphisms, natural transformations, Yoneda’s lemma, representable functors, adjoint functors, equivalences of categories, limits and colimits (e.g., for limits: products, equalizers, pullbacks, initial objects).

Homological Algebra: exact sequences, splittings, snake and five lemmas, projective, injective, and flat modules, complexes, (co)homology.

Commutative Ring Theory: localizations, Hilbert’s basis theorem, integral extensions, radicals of ideals, Zariski topology and Hilbert’s Nullstellensatz, Dedekind rings, DVRs.

Noncommutative Ring Theory: Associative and graded algebras, endomorphism rings, group rings, semisimple rings, irreducible and indecomposable modules, central simple algebras, Artin-Wedderburn theorem, Jacobson radicals.

Representation Theory: basic definitions and examples (e.g,. representations, regular and trivial representations), Schur’s lemma, Maschke’s theorem, characters of finite groups, class functions, orthogonality relations, character tables, induced characters.

References

• Dummit and Foote, Abstract Algebra.

• Grillet, Abstract Algebra.

• Hungerford, Algebra.

• Jacobson, Basic Algebra I and II.

• Lang, Algebra.