  The Binomial Theorem has long been essential in mathematics. In one form or
another it was known to the ancients and, in the hands of Leibniz, Newton,
Euler, Galois, and others, it became an essential tool in both algebra and
analysis. Indeed, Newton early on developed certain binomial series (see
Section \ref{newton}) which played a role in his subsequent work on the
calculus. From the work of Leibniz, Galois, Frobenius, and many others, we know
of its essential role in algebra. In this paper we rapidly trace the history of
the Binomial Theorem, binomial series, and binomial coefficients, with emphasis
on their decisive role in function field arithmetic. We also explain conversely
how function field arithmetic is now leading to new results in the binomial
theory via insights into characteristic $p$ $L$-series.

Number Theory