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. ...