Course Details
Subject {L-T-P / C} : MA6612 : Algebra { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Ranjit Mehatari
Syllabus
Introduction to group theory, Group Action, Fixed Sets and Isotropy Groups, Orbits, Class equation of an action, p-Groups, Sylow Theorems, Subnormal and Normal Series, Schreier’s Theorem, Composition Series, Jordan-Holder Theorem , Solvable Groups, Nilpotent Groups, Introduction to ring theory, Factorization in Rings, Euclidean Domain, Principal Ideal Domain, Unique Factorization Domain, Gauss Theorem, Einstein’s Irreducibility Criterion, Chinese Remainder Theorem, Field extension, Algebraic and Transcendental elements, Finite and Algebraic Extensions, Geometric Constructions, Splitting Field and Normal Extension, Separable Polynomial and Separable Extension, Perfect field, Galois Extension, Fundamental Theorem of Galois Theory, Cyclotomic extensions, Norm, trace and discriminant, Solvability of Polynomial equations by radicals, Cyclic extensions, Abelian extensions, Transcendental extensions.
Course Objectives
- To get a broad idea of groups, rings and fields.
- To understand use of ideals in a commutative rings.
- To introduce the notion of field extensions and their applications.
Course Outcomes
Good knowledge in abstract algebra. They will have an abstract visualization rings and fields. They will be able to construct finite fields. They will be able to understand meaning of an field extension. By studying this course they will be able to solve exercise problems by themselves.
Essential Reading
- I. N. Herstein, Abstract Algebra, 3rd Edition, John Wiley and Sons
- J. A. Gallian, Introduction to commutative algebra, Narosa
Supplementary Reading
- D. S. Dummit & R. M. Foote, Abstract Algebra, John Wiley and Sons
- J. J. Rotman, An Introduction to the Theory of Groups, Springer