Algebra (Ph.D. and Ed.D.)

Preparatory Courses: Math 5613, 5623

1.

Group Theory: Homomorphism theorems; direct products; symmetric groups; normal subgroups; Sylow theorems; abelian groups; free groups; solvable groups; nilpotent groups; generators and relations

2.

Ring Theory: Polynomial rings; PIDs; Euclidean domains; FDs; fields of fractions; Noetherian and Artinian rings; radicals; prime and maximal ideals; semisimple rings; primitive rings; Hilbert basis theorem

3.

Module Theory: Projectives; injectives; free modules; tensor products; fundamental theorem on finitely generated modules over PIDs; exact sequences

4.

Field Theory: Extensions; splitting fields; separability; Galois theory; finite fields; Fundamental Theorem of Algebra

REFERENCES: S. Lang, Algebra; T. Hungerford, Algebra ; Herstein, Topics in Algebra; E. Artin, Galois Theory; L. Grove, Algebra.