Algebra 2

This is a continuation of the first course of algebra (Algebra 1). It starts from the second isomorphism theorem, and then proceeds by discussing basic classes of groups and rings including in particular symetric, soluble, abelian free groups and Noetherian and Dedekind rings.

Divisiblility theory in a polynomial ring, splitting field of a polynonial and the algebraic closure of a field are also discussed.

After completing the course a student should know the following topics and have proficiency in their applications:

The second isomorphism theorem for groups.

Free abelian groups (sets of free generators, rank, subgroups).

The structure of the finitely generated abelian groups.

Symmetric groups (normal subgroups of Sn, simplicity of An).

Soluble groups (composition sequences, Jordana-H?lder theorem).

Noetherian rings (the Hilbert basis theorem).

Dedekind domains (uniform factorization into prime ideals).

Divisibility in polynomial rings (Gauss lemma, unique factorization polynomial rings, Eisenstein criterion).

Splitting field of a polynomial (existence and uniqueness).

Algebraic closure of a field (existence and uniqueness).

S. Lang, Algebra, Addison-Wesley, 1965.

T. W. Hungerford, Algebra, Springer 1979.