# Algebra

### Algebra Syllabus

For the Oral Qualifying Exam

#### Groups

• Elementary concepts (homomorphism, subgroup, coset, normal subgroup), 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).
• Integral domains, Euclidean domains, principal ideal domains, polynomial rings, Eisenstein's irreduciblility criterion, Chinese remainder theorem.
• Structure of finitely generated modules over a prinicpal 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 extensions and solving equations by radicals.
• Finite fields.

### References

• M. Artin, Algebra
• Dummit and Foote, Algebra
• S. Lang, Algebra
• T.W. Hungerford, Algebra