Algebra qualifier topics
Last revised: April 2010
Sample references:
P. Aluffi, Algebra: Chapter 0, AMS.
M. Artin, Algebra, Prentice Hall.
D. Dummit, R. Foote, Abstract Algebra, Wiley.
S. Lang, Algebra, Springer Verlag.
Basic topics:
Categories: Definition; universal properties.
Groups:
Basic definitions; subgroups; cosets; normal subgroups; quotient
groups; homomorphisms; fundamental theorems on group homomorphisms;
free groups; permutation groups; Lagrange and Cayley's theorem;
actions of a group on a set; class formula; Sylow's theorems;
composition series and solvability; products and semidirect
products; classification of finite abelian groups.
Rings:
Basic definitions; ideals; quotient rings; ring homomorphisms;
fundamental theorems on ring homomorphisms; fields; prime and
maximal ideals; Noetherian rings; Hilbert's basis theorem;
irreducibility; UFDs, PIDs, Euclidean domains; Gauss's lemma;
Chinese Remainder Theorem.
Modules:
Basic definitions; isomorphism theorems; free modules and algebras;
Noetherian modules; complexes and homology; exact sequences, the snake
lemma; free modules; presentations and resolutions; classification of
finitely generated modules over PIDs.
Linear algebra:
Vector spaces, basic definitions; linear independence, bases; matrices;
Gaussian elimination; determinants and non-singularity;
eigenvectors and eigenvalues; minimal and characteristic polynomials;
Cayley-Hamilton theorem; Rational and Jordan canonical forms.
