Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Dr. A. Satyanarayana Reddy

Department of Mathematics,
Shiv Nadar University

6

Field Extensions, Algebraic Extensions, Splitting Fields

Topics: Fields, prime subfields, Field extensions, degree, tower theorem, Algebraic and transcendental elements / extensions, Monomorphisms of field extensions, Splitting fields, Uniqueness of splitting field Normal extensions, Introduction to cyclotomic fields, Ruler and compass constructions (brief, if time permits)

Tutorial 1: Problems on degree of extensions, tower theorem, finding splitting fields, constructing monomorphisms.

Suggested Texts:

• Garling, Chapters 1–5;

• Dummit & Foote, Sections 13.1–13.4,

• M. Artin:Algebra, 2nd Edition, Prentice Hall of India, 2011.

Dr. Anuj Bishnoi,

Department of Mathematics, University of Delhi

6

Galois Extensions, Fundamental Theorem of Galois Theory

Topics Covered: Separability, separable extensions, Automorphism groups, Galois groups, Fixed fields, Galois extensions, Fundamental theorem of Galois theory,  Galois theory of polynomials, Solution by radicals (if time permits)

Tutorial 2: Normal and separable extensions, computation of Galois groups

Tutorial 3: Applications of the Fundamental Theorem (correspondence, intermediate fields)

Suggested Texts:

• Garling, Chapters 6–9;

• Dummit & Foote, Sections 13.5, 14.1–14.2,

• M. Artin: Algebra, 2nd Edition, Prentice Hall of India, 2011.

6

Basics of normed spaces

Basics aspects of normed spaces, Some examples, Finite dimensional normed spaces and local compactness, Local compactness lost in infinite dimensional normed spaces, Separability in normed spaces

Suggested Texts:

• G. Bachman and L. Narici, Functional Analysis

• E. Kryeszig, Functional Analysis

• G. F. Simmons, Topology and Modern Analysis

6

Fundamental theorems, Hilbert spaces and bounded operators

Operators between normed spaces, Hahn-Banach Theorem (statement) and applications, Some category theorems (statements) and applications, Hilbert spaces, Riesz representation theorem, various types of operators between Hilbert spaces

Suggested Texts:

• G. Bachman and L. Narici, Functional Analysis

• E. Kryeszig, Functional Analysis

• G. F. Simmons, Topology and Modern Analysis

S. No.

Name

Affiliation

1

(For Functional Analysis)

Dr. Rachna Aggarwal

Department of Mathematics,
Amity University, Mohali

Dr. Ashutosh Pandey

Department of Applied Sciences and Humanities,
Faculty of Technology, University of Delhi

2

(For Galois Theory)

Dr. Manisha Saini

Department of Mathematics,
Hansraj College, University of Delhi

Dr. Ashima

Department of Mathematics,
Hansraj College, University of Delhi