## 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