USRA project: Central-simple algebras and Galois cohomology

One learns in MAT3143 that there exists rings that behave like fields (every non-zero element is invertible), except that their multiplication is not commutative, the so-called quaternions. In MAT3143 they are usually defined as a certain subspace of complex 2 x 2 matrices or as a 4-dimensional real vector space. In any case, it is an important fact for modern algebra that quaternions exist in much more generality, over any field, even over any ring. One way to construct them is to use another fundamental method of modern algebra: Galois descent and Galois cohomology, related to Galois extension of fields. The aim of this USRA project is to introduce the student to the techniques and fundament results of generalized quaternions, called central-simple algebras and the methods required to study them.

Topics to be covered are: Quaternion algebras, general central-simple algebras, Wedderburn's Theorem, Galois descent, Brauer group. This will cover about the first 3 chapters of the book by Gille-Szamuely on "Central-simple algebras and Galois cohomology". These are topics usually not covered in a standard undergraduate program, yet they are fundamental for research in algebra. The research component of the project will consist in solving some of the exercises in the textbook and in presenting summaries of sections in the textbook.

Prerequisites: a third-year course in basic ring theory and some advanced linear algebra. The project will be an excellent preparation for a MSc thesis.