Groups, Rings and Number Theory (2022)

School Convener(s)

1. Polynomials and Primitive Roots: Background and Motivation; Examples, Discrete Log
2. Existence of Primitive Roots; Proofs and applications.
3. Power Residues and Quadratic Resides.
4. Quadratic Reciprocity Law: Proof and applications.

1. Recalling basic linear algebra, basic group actions, inner product spaces and infinite groups GL(n) and SU(n)..
2. Definition and examples of permutation representations of finite groups.
3. Definition and examples of complex representations of finite groups.
4. Notion of Equivalence of representations, irreducibility and complete reducibility.
5. Proof of Maschke's theorem over C using inner products.
6. Definition of projection maps and proof of Maschke for more general fields.
7. Schur's lemma and Schur's first and second orthogonality relations.
8. Brief description of finite-dimensional representations of SU(2).

1. Number fields, algebraic integers and integral extensions, trace and norms.
2. Quadratic number fields, ring of integers QK in a quadratic number field K = Q[√d], units in ܳܭ for d < 0. Ideals, product of ideals, prime and maximal ideals in arbitrary (commutative) rings (with identity).
3. Lattices in C, lattice basis, similarity and ideal classes. Unique factorisation of ideals in ܳܭ for K imaginary quadratic number field.