Course Details
Subject {L-T-P / C} : MA4104 : Algebra: Ring and Field Theory { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Bikramaditya Sahu
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.
- To understand solvability of certain polynomials using Galois theory.
Course Outcomes
They will have an abstract visualisation of groups, rings and fields. They will be able to construct finite fields and also will be able to understand meaning of a field extension. By studying this course they will be able to solve exercise problems by themselves. This course will help students to do research in related subjects.
Essential Reading
- I. N. Herstein, Abstract Algebra, 3rd Edition, John Wiley and Sons, 2023
- J. A. Gallian, Contemporary Abstract Algebra, 4th Edition, Narosa, 2021
Supplementary Reading
- D. S. Dummit & R. M. Foote, Abstract Algebra, 3rd Edition, John Wiley and Sons, 2011
- J. J. Rotman, An Introduction to the Theory of Groups, 4th Edition, Springer, 2014