TEW - Groups, Rings and Number Theory (2022)

Speakers and Syllabus

Detailed Syllabus

1. Syllabus to be covered by B. Sury: Group Representations
   1. Basic definitions and first examples
   2. Irreducible Representations
   3. Unitary Representations
   4. Maschke’s theorem and complete reducibility
   5. Character Theory and the Orthogonality Relations
   6. Schur’s Lemma
   7. Proof of the Orthogonality Relations
   8. Representations of SU2
   9. Reference books: Chapter 10 of Algebra by M Artin and Representation Theory of Finite Groups by B. Steinberg
2. Syllabus to be covered by S. Laishram: Primitive Roots and Quadratic Reciprocity Law
   1.  Motivation
   2.  Polynomials and Primitive Roots
   3.  Existence of primitive roots
   4.  Primitive Roots to Composite Moduli
   5.  Explicit description of the group U (Z/nZ) for arbitrary n.
   6.  Power Residues and Quadratic Residues
   7.  Gauss’s Lemma and Quadratic Reciprocity Law
   8.  Applications
   9. Reference book: Chapter 4 and 5 of A Classical Introduction to Modern Number Theory by K. Ireland and M. Rosen; Chapter 3 of Elementary Methods in Number Theory by M. Nathanson.
3. Syllabus to be covered by R. Mawia: Quadratic Number Fields
   1. Algebraic Integers
   2. Factoring Algebraic Integers
   3. Ideal Multiplication and Factoring Ideals
   4. Prime Ideals and Prime Integers
   5. Ideal Classes
   6. Computing the Class Group
   7. Real Quadratic Fields
   8. Reference book: Chapter 13 of Algebra by M Artin.

Time Table

Time Table