Unit 1

Review of Groups, Normal subgroups, Factor or Quotient Groups and Homomorphisms. Isomorphism of Groups, Cayley’s Theorem. External Direct Products, Fundamental theorem of Finite Abelian Groups. Groups acting on themselves by Conjugation- Class Equation, Sylow’s Theorem and its applications, Simple Groups, Non Simplicity Tests.

Unit 2

Review of Rings: Integral domains, Quotient Ring and Ideals, Properties of ideals, Prime and Maximal ideals, Chinese remainder theorem, Ring homomorphisms, Polynomial rings, Polynomial rings over Fields, Division algorithm.

Unit 3

Principal ideal domain, Factorisation of Polynomials, Gauss’s lemma, Eisenstein’s irreducibility criteria, Unique Factorisation in Z[x],Euclidean domain, Unique factorization domain.

Unit 4

Field extensions, Fundamental theorem of Field theory, Splitting fields, Zeros of an irreducible polynomial, Algebraic and Transcendental extensions, Finite extensions, Properties of Algebraic extensions.

Unit 5

Finite fields, Classification of Finite fields, Structure of Finite fields, Subfields of a Finite field. Galois Theory: Automorphims of fields, Galois group, examples, Fundamental theorem of Galois Theory and its applications, Cyclotomic extensions, Solving of Polynomials by Radicals, Insolvability of Quintic.

Course outcomes
CO 1 Understand the familiarity with the concepts of ring and field, and their main algebraic properties;
CO 2 Understand correctly use the terminology and underlying concepts of Galois Theory in a problem-solving context
CO 3 Ability to Reproduce the proofs of its main theorems and apply the key ideas in similar arguments;
CO 4 Ability to Calculate Galois groups in simple cases and to apply the group-theoretic information to deduce results about fields and polynomials
CO 5 Ability to Demonstrate the capacity for mathematical reasoning through analyzing, proving and explaining concepts from field extensions and Galois Theory and apply problem –solving in diverse situation in physics, engineering and other mathematical contexts.

Textbook

1. Joseph A. Gallian, Contemporary Abstract Algebra, 4th edition, Narosa.

References

• Dummit and Foote, Abstract Algebra, Wiley.
• M. Artin, Algebra, Prentice Hall. I.N.Herstein, Topics in Algebra, Wiley.
• N. Jacobson, Basic Algebra 1 & 2, Dover.