Material for Preliminary Examination B -- Algebra Section

(Revised Dec 11, 2007)

The following list gives topics on which the Preliminary Examination B in Algebra will be based. Math 5331 covers many, but not necessarily all, of these topics. Students (even those who have taken Math 5331) are advised to prepare for the examination using many resources, including (but not limited to) the books suggested below.

1. Groups

(a) definition, properties, examples

(b) Lagrange’s Theorem

(c) normal subgroups, kernel, image, quotient groups, homomorphism/isomorphism theorems

(d) cyclic groups, abelian groups, matrix groups, dihedral groups, symmetric groups

(e) centralizers, normalizers, stabilizers

(f) fundamental theorem of finitely generated abelian groups

(g) simple groups, normal series, Jordan-Hӧlder theorem

(h) solvable groups, unsolvability of Sn for n > 4, nilpotent groups

(i) group actions, class equation, permutation representations, Cayley’s Theorem

(j) Sylow’s Theorems

(k) solvability of p-groups

(l) free groups

(m) direct products

2. Rings

(a) definition, properties, examples

(b) left, right and two-sided ideals and their properties

(c) quotient rings, homomorphism/isomorphism theorems

(d) matrix rings, group rings, polynomial rings, real quaternions

(e) integral domains, fields

(f) noncommutative domains, division rings

(g) prime ideals, maximal ideals, local rings

(h) localization in commutative rings: multiplicative sets, rings of fractions

(i) products

(j) Chinese Remainder Theorem for commutative rings

(k) Euclidean rings, PIDs, UFDs, application to polynomial rings, Gauss’ Lemma

(l) proof that a polynomial ring over a UFD is a UFD

3. Modules

(a) definition, properties, examples

(b) submodules, quotient modules, homomorphism/isomorphism theorems, cyclic modules, simple modules

(c) products, direct sums (internal and external)

(d) free modules

(e) applications to abelian groups