This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
Prerequisites
Students who take Math 601 should have taken a semester long course in abstract algebra (e.g. Math 501 at Duke) and should be familiar with the following topics: linear algebra, equivalence relations, equivalence classes, the rudiments of group theory including subgroups, homomorphisms, group actions; the bare rudiments of ring theory including ring homomorphisms and ideals.
Syllabus
- Groups: Elementary concepts (homomorphism, subgroup, coset, normal subgroup, simple group). Solvable groups, commutator subgroup, Sylow theorems, structure of finitely generated Abelian groups. Symmetric, alternating, dihedral, and general linear groups.
- Rings: Commutative rings and ideals (principal, prime, maximal). Chinese remainder theorem. Integral domains, factorization, principal ideal domains, Euclidean domains, polynomial rings, Gauss Lemma,Eisenstein's irreducibility criterion.
- Modules: Elementary concepts: homomorphism, linear independence, exact sequence, finite presentation, torsion. Structure of finitely generated modules over a principal ideal domain.
- Fields: Extensions: finite, algebraic, separable, inseparable, transcendental, splitting field of a polynomial, primitive element theorem, algebraic closure. Finite Galois extensions and the Galois correspondence between subgroups of the Galois group and subextensions. Solvable Galois groups and the problem of expressing the roots of a polynomial in terms of radicals. Finite fields.
References
- Abstract Algebra by Dummitt and Foote
- Algebra by Michael Artin
- Algebra by Serge Lang
- Algebra by T.W. Hungerford