Total credits: 4 (3 Lectures + 1 Tutorial)
Pre-requisite of the course: Basics of Groups and Rings
Course Objectives: The primary objective of this course is to:
Study the ancient problem of solvability of polynomials over the rational field.
Understand a necessary and sufficient condition for the solvability of polynomials in terms of radical expressions.
Apply Galois theory to classical problems, such as the insolvability of the general quintic.
Course Learning Outcomes: This course will enable the students to:
Identify and construct examples of fields, distinguish between algebraic and transcendental extensions, characterize normal extensions in terms of splitting fields and prove the existence of algebraic closure of a field.
Characterize perfect fields using separable extensions, construct examples of automorphism group of a field and Galois extensions as well as prove Artin’s theorem and the fundamental theorem of Galois theory.
Classify finite fields using roots of unity and Galois theory and prove that every finite separable extension is simple.
Use Galois theory of equations to prove that a polynomial equation over a field of characteristic is solvable by radicals if and only if its group (Galois) is a solvable group and hence deduce that a general quintic equation is not solvable by radicals.
Contents: Unit I: Fields and their extensions, Splitting fields, Normal extensions, Algebraic closure of a field.
Unit II: Separability, Perfect fields, Automorphisms of field extensions, Artin’s theorem, Galois extensions, Fundamental theorem of Galois theory.
Unit III: Roots of unity, Cyclotomic polynomials and extensions, Finite fields, Theorem of primitive element, and Steinitz’s theorem.
Unit IV: Galois theory of equations, Theorem on natural irrationalities, Radical extension and solvability by radicals.
Suggested Readings:
[1] P. M. Cohn, Basic Algebra, Springer International Edition, 2003
[2] D. S. Dummit & R. S. Foote, Abstract Algebra, Wiley Student Edition, 2011
[3] T. W. Hungerford, Algebra, Springer-Verlag, 1981.
[4] N. Jacobson, Basic Algebra, Volume I, Dover Publications Inc., 2009.
[5] I. Stwart, Galois Theory, CRC Press, 2015.
Tutorial sheets:
Total credits: 4 (3 Lectures + 1 Tutorial)
Pre-requisite of the course: Knowledge of metric spaces.
Course Objectives: The primary objective of this course is to introduce:
Basic principles of point-set topology, including bases and subbases for a topology.
Continuity, homeomorphisms, and different types of topologies, such as product and box topologies.
Key notions of connectedness and local connectedness.
Compactness and its significance in topological spaces.
Course Learning Outcomes: This course will enable the students to:
Analyze subsets of topological spaces by determining their interior, closure, boundary, and limit points, as well as identifying bases and subbases.
Identify continuous functions between topological spaces, analyze mappings into product spaces, and compare topological properties of different spaces .
Evaluate the connectedness and path connectedness of the product of an arbitrary family of spaces.
Understand key classifications of topological spaces, including Hausdorff spaces, first and second countable spaces, and separable spaces.
Explore advanced concepts such as limit point compactness and Tychonoff’s theorem.
Contents: Unit I: Topological spaces, Basis, Order topology, Subspace topology, Metric topology, Closed set and limit points, Hausdorff spaces.
Unit II: Continuous functions, Homeomorphism, The box and product topologies, Metrizability of products of metric spaces, Connected and path connected spaces.
Unit III: Locally connected and locally path connected spaces, Connectedness of product of spaces, First and second countable spaces, Separable spaces.
Unit IV: Compact spaces, The Tychonoff theorem, Limit point compactness, Sequential compactness.
Suggested Readings:
[1] J. R. Munkres, Topology, Updated Second Edition, Pearson, 2021.
[2] T. B. Singh, Introduction to Topology, Springer Nature, 2019.
[3] G. E. Bredon, Topology and Geometry, Springer, 2014.
[4] J Dugundji, Topology, Allyn and Bacon Inc., Boston, 1978.
[5] J L. Kelley, General Topology, Dover Publications, 2017.
Tutorial sheets:
Total Marks: 100 (70 End Semester Examination + 30 Internal Assessment)
Course Objectives: Introduce the notion of homotopy, groups with pointed spaces and covering spaces which is closely associated with the fundamental groups. Moreover, to introduce the concept of free groups and presentation of a group and to compute the fundamental group of wedge of circles, Klein bottle, adjunction of a disc and a path connected space.
Course Learning Outcomes: After studying this course the student will be able to
CO1. Grasp the basics of Algebraic Topology.
CO2. Determine fundamental groups of some standard spaces like Euclidean spaces and spheres.
CO3. Understand proofs of some beautiful results such as Fundamental theorem of Algebra, Brower’s fixed-point theorem, BorsukUlam theorem.
Contents: Unit I: Homotopic maps, homotopy type, retract and deformation retract.
Unit II: Fundamental group, Calculation of fundamental groups of n-sphere, cylinder, torus, and punctured plane, Brouwer’s fixed-point theorem, Fundamental theorem of Algebra.
Unit III: Free products, Free groups, Seifert-Van Kampen theorem and its applications.
Unit IV: Covering projections, Lifting theorems, Relations with the fundamental group, Universal covering space, BorsukUlam theorem, Classification of covering spaces.
Suggested Readings:
[1] G. E. Bredon, Topology and Geometry, Springer, 2014.
[2] W. S. Massey, A Basic Course in Algebraic Topology, World Publishing Corporation, 2009.
[3] J. J. Rotman, An Introduction to Algebraic Topology, Springer, 2011.
[4] T. B. Singh, Elements of Topology, CRC Press, Taylor & Francis, 2013.
[5] E. H. Spanier, Algebraic Topology, Springer-Verlag, 1989.
Tutorial sheets:
Previously taught courses:
M.Sc. Mathematics:
MMATH18-101: Field Theory
MMATH18-201: Module Theory
MMATH18-202: Introduction to Topology
MMATH18-301(i): Algebraic Topology
MMATH18-401(i): Advanced Group Theory
MMATH18-401(iii): Simplicial Homology Theory
MMATH18-403(ii): Differential Geometry
B.Sc. Mathematics:
List of courses taught at Satyavati College, Kirori Mal College and Keshav Mahavidyalaya at B.Sc. level:
Calculus, Algebra, Analysis, Differential Equations, Mechanics, Probability theory, Statistics, Multivariate calculus, etc.