The Ongoing Binomial Revolution

Abstract

The Binomial Theorem has played a crucial role in the development of mathematics, algebraic or analytic, pure or applied. It was very important in the development of calculus, in a variety of ways, and has certainly been as important in the development of number theory. It plays a dominant role in function field arithmetic. In fact, it almost appears as if function field arithmetic (and a large chunk of arithmetic in general) is but a commentary on this amazing result. In turn, function field arithmetic has recently returned the favor by shedding new light on the Binomial Theorem. It is our purpose here to recall the history of the Binomial Theorem, with an eye on applications in characteristic p, and finish by discussing these new results.