Math 500. Abstract Algebra I

Textbook: Up to the discretion of the advisor. Rotman’s Advanced Modern Algebra is suitable. Hungerford’s or Lang’s Algebra are other options. Additional reading material for certain topics is indicated below.

(1) Group Theory

(a) Isomorphism theorems, factorization of homomorphisms using diagrams.
(b) G-sets. Transitivity, orbits, stabilizers. Groups acting on coset spaces. Conjugacy classes. Normalizers, centralizers, quotient groups, products of groups.
(c) Symmetric groups, alternating groups.
(d) Subnormal and normal series, solvable groups.
(e) Sylow’s theorems.
(f) Examples of groups of small order.
(g) Linear groups: Classical linear groups, example of SU2 and SL2, oneparameter subgroups and the Lie algebra, simple groups. [Material of this and the next topic is covered in e.g. Artin’s Algebra Ch 8, 9.]
(h) Group representations: Definition, G-invariant forms, irreducible representations, Schur’s Lemma, character theory for finite groups.

(a) Algebraic and transcendental extensions, existence and uniqueness of algebraic closures.
(b) Separable and normal extensions. Extensions of automorphisms. Galois groups as permutation groups.
(c) Splitting fields. Characterization of Galois extensions. Finite fields.
(d) Fundamental theorem of Galois theory. Fundamental theorem of symmetric functions.
(e) Examples of computations of Galois groups. Cyclic extensions.
(f) Cyclotomic extensions. Irreducibility of cyclotomic polynomial. Cyclicity of finite multiplicative groups in fields. Radical extensions.
(g) Characterization of solvability by radicals.

(a) Prime ideals. Maximal ideals. Examples of Euclidean domains and principal ideal domains.
(b) A PID is a UFD. Gauss’s Lemma.
(c) Structure theorem for modules over a PID.
(d) Application to abelian groups.