Page 4 - DMTH403_ABSTRACT_ALGEBRA

SYLLABUS

Abstract Algebra
Objectives:
To learn about the structure as group, ring and field.

To gain knowledge about homomorphisms, isomorphisms, cosets, quotient groups, and the isomorphism theorems,

rings, ideals, ring homeomorphisms, isomorphisms and its theorems.
To learn about fields, quotient fields and field extensions Galois Theory also.


Sr. No. Content
1 Groups : Definition and examples, Quotient groups, Cyclic groups, Permutation

groups and The alternating groups, Subgroups, normal subgroups and the

commutator subgroup, Generating sets, Lagrange's Theorem and Cayley's
theorem

2 Homomorphisms and Automorphisms, Direct products. External and internal
direct products,

3 Structure of finite abelian groups, Conjugate elements and class equations of
finite groups, Sylow's theorems and their simple applications.

4 Solvable groups,Jordan-Holder Theorem, Rings, Subrings, Ideals and their

operations
5 Factor rings and Homomorphisms, Integral domains

6 Polynomial rings,The field of quotients Euclidean domains, Principal Ideal

Domains, Unique factorization domain
7 Prime fields, finite and algebraic extensions, Roots of a polynomial

8 splitting fields; existence and uniqueness, Separable extensions, Finite fields; the

structure, the existence of GF (pn)
9 Galois theory :Normal extensions, Galois groups

10 Symmetric functions, fundamental theorem, Constructible polygons, Solvability

by radicals