Group Theory-I: Course Outline algebra II

Rings:  

Definition,  examples.

Quadratic  integer  rings.  

Examples of non-commutative rings.

The Hamilton quaternions.

Polynomial rings.

Matrix rings.

Units, zero-divisors, nilpotents, idempotents.

Subrings, Ideals.

Maximal and prime Ideals.

Left, right and two-sided ideals;.

Operations with ideals.

The ideal generated by a set.  

Quotient rings. Ring   homomorphism.  

The   isomorphism   theorems,   applications.

Finitely generated ideals.

Rings of fractions.

Integral  Domain:  

The  Chinese  remainder  theorem.  

Divisibility  in integral domains, greatest common divisor, least common multiple.

Euclidean domains.

The Euclidean algorithm.

Principal ideal domains.

Prime and irreducible elements in an integral domain.

Gauss lemma, irreducibility criteria  for  polynomials.

Unique  factorization  domains.

Finite fields.

Polynomials in several variables.

Symmetric polynomials.

The fundamental theorem of symmetric polynomials.