Syllabus for the Algebra Qualifying Exam
Recommended book: Algebra, Lang (revised third edition)
Groups (Ch. 1)
Isomorphism theorems; permutation groups; group actions; p-groups and Sylow’s theorem; solvable groups; composition series; Jordan-Holder theorem.
Rings, Modules and Commutative Algebra (Ch. 2,3 and 4)
Ideals; isomorphism theorems; prime and maximal ideals; radicals; Chinese remainder theorem; polynomial and power series rings; PIDs, UFDs, Gauss’s lemma, Eisenstein’s criterion. Cyclic modules; structure theorem of finitely generated modules over PIDs.
Nakayama lemma; Noetherian rings; Hilbert basis theorem; localization. Hilbert’s Nullstellensatz; integral extensions; Noether normalization; exact sequences; tensor products and multilinear algebra; free, flat module. Prime spectrum of rings; Zariski topology.
Field Theory (Ch. 5 and 6)
Finite, algebraic, separable, normal extensions; minimal polynomial; primitive element theorem; splitting field; algebraic closure. Finite fields: classification; Frobenius automorphism, cyclicity multiplicative subgroup. Galois extension; Galois groups; fundamental theorem of Galois theory; solvability by radicals via solvable groups.
Linear Algebra (Ch. 13,14,15 and 16)
Rational and Jordan canonical form, minimal and characteristic polynomials, traces and determinants, eigenspaces and generalized eigenspaces, diagonalization, commuting matrices. Bilinear forms: orthogonal, symplectic, unitary groups. Spectral theorem over R,C.
Representation Theory and Non-Commutative Algebra (Ch. 17 and 18)
Linear representations of finite groups; Schur’s lemma; characters; orthogonality relations, tensor/dual.
Simple and semi-simple rings and modules, Artin–Wedderburn theorem.
Homological Algebra (Ch. 20)
Categories; morphisms; functors; natural transformations; equivalence of categories; limit and colimit; exactness in abelian categories, projective and injective modules; resolutions; Ext and Tor