Title of the course: Quadratic Forms and the Hasse-Minkowski Theorem
Instructor: Dr. Martin Djukanovic
Institution: Ulm University
Dates: 6-12 August 2018
Prerequisites: Basic algebra, basic analysis helps but is not necessary.
Level: Graduate, advanced undergraduate
Abstract:We recall the concept of a Diophantine equation and some classical results and examples. We introduce the Legendre symbol and prove the Quadratic Reciprocity Law. Then we introduce p-adic numbers and prove Hensel's Lemma. We examine ternary quadratic forms over the p-adics and prove the basic properties of the Hilbert symbol. Finally, if time permits, we state and prove the Hasse-Minkowski theorem for arbitrary quadratic forms over the rationals.