Choosing elements from finite fields

Michael Vaughan-Lee

Published 2017-07-30Version 1

In two important papers from 1960 Graham Higman introduced the notion of PORC functions, and he proved that for any given positive integer $n$ the number of $p$-class two groups of order $p^n$ is a PORC function of $p$. A key result in his proof of this theorem is the following: "The number of ways of choosing a finite number of elements from the finite field of order $q^n$ subject to a finite number of monomial equations and inequalities between them and their conjugates over GF($q$), considered as a function of $q$, is PORC." Higman's proof of this result involves five pages of homological algebra. Here we give a short elementary proof of the result. Our proof is constructive, and gives an algorithms for computing the relevant PORC functions.