finite field

A Combinatorial Identity Using Finite Fields

Achyut Bharadwaj
September 2022
finite field, automorphism

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. ...

Cardinality of Finite Fields

Achyut Bharadwaj
September 2022
cardinality, finite field, automorphism, characteristic, isomorphism, homomorphism, Fermat's Little Theorem, FLT

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. ...