MATH 7390: Topics in Algebra: Introduction to p-adic Numbers (Fall 2011)

Instructor: Shankar Sen

This will be an introduction to p-adic numbers, more or less from scratch, along with the classical “application” to quadratic forms. The latter is the Hasse-Minkowski theory which solves the following two problems:

1. Given an equation f(X₁,...,X_n) = 0 where f is a quadratic form, in n variables, with rational coefficients, decide if there is a non-trivial solution over the rational numbers.
2. Find invariants of forms as in (1) which will determine if two such forms are equivalent, i.e., if one can be transformed into the other by a linear change of variables over the rational numbers.

This theory is by now a bit over a century old, but still serves as a model for the modern attacks on the very much harder questions on Diophantine equations of higher degree.

Requirements for the course: an undergraduate course in modern algebra at the level of MATH 4340. It will be accessible to advanced undergraduates and beginning graduate students.