Choosing elements from finite fields

TL;DR

This paper provides an elementary, constructive proof and algorithms for computing PORC functions related to choosing elements from finite fields, simplifying Higman's original homological algebra approach.

Contribution

It offers a simplified, elementary proof and practical algorithms for PORC functions in finite field element selection, improving understanding and computation.

Findings

Elementary proof of Higman's PORC function result

Algorithms for computing PORC functions

Simplification of homological algebra methods

Abstract

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.