A Combinatorial Identity Using Finite Fields

September 2022

Introduction #

Consider a prime $p$. For what integers $n$ does $p$ divide all of $$\binom{n}{1}, \binom n 2, \binom n 3, \dots, \binom{n}{n-1}?$$ Can we characterize all such $n$ given a value of $p$? It turns out that this happens if and only if $n$ is a perfect power of $p$. How do we prove this? In fact, there exists a simple proof using elementary methods. But as always, it is both fun as well as good to prove everything twice. In this article I present an interesting way to characterize all such $n$ using concepts of finite fields.

Cardinality of Finite Fields

September 2022

Introduction #

A well known theorem about finite fields states the following.

$F$ is a finite field if and only if $|F| = p^k$ for some prime $p$ and positive integer $k$.

In this write-up, we prove the first part of the above theorem, i.e. if $F$ is a finite field, then $|F| = p^k$ using a different approach.

Introduction to Finite Fields #

We first list down some basic definitions that will be used as we move forward in this write-up.