Algebra Qualifying Syllabus

Group Theory: Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and alternating groups; free groups, structure theorem for finite abelian groups.

Group Actions: Groups acting on sets, cosets, and themselves; orbits and stabilizers, permutation representations, Cayley’s Theorem, the class equation, inner automorphisms and automorphism groups, p-subgroups and the Sylow Theorems.

Ring Theory: Definition and examples, homomorphisms, ideals, quotients, integral domains and their fields of fractions, maximal and prime ideals. Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains, Principal Ideal Domains, Gauss’s Lemma, polynomial factorization, Eisenstein’s criterion, Gaussian integers.

Modules: Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications to linear operators: Jordon and rational canonical forms.

Basic Linear Algebra: vector spaces, bases, dimension, bases for infinite dimensional spaces (Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem.

Field Theory: Definition and examples, algebraic and transcendental extensions, degree of a finite extension, multiplicativity of degrees, adjunction of roots, finite fields.

Basic Galois Theory for Finite Separable Extensions: Definitions of Galois group and Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer extensions, cyclotomic extensions, quintic polynomials.