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1 Introduction to StandardFF package

1.1 Aim

This GAP-package provides a reference implementation for the standardized constructions of finite fields and generators of cyclic subgroups defined in the article [Lüb22].

The main functions are FF (2.2-1) to construct the finite field of order p^n and StandardCyclicGenerator (3.1-1) to construct a standardized generator of the multiplicative subgroup of a given order m in such a finite field. The condition on m is that it divides p^n-1 and that GAP can factorize this number. (The factorization of the multiplicative group order p^n-1 is not needed.)

Each field of order p^n comes with a natural F_p-basis which is a subset of the natural basis of each extension field of order p^nm. The union of these bases is a basis of the algebraic closure of F_p. Each element of the algebraic closure can be identified by its degree d over its prime field and a number 0 ≤ k ≤ p^d-1 (see SteinitzPair (2.4-1)) or, equivalently, by a certain multivariate polynomial (see AsPolynomial (2.3-1)). This can be useful for transferring finite field elements between programs which use the same construction of finite fields.

The standardized generators of multiplicative cyclic groups have a nice compatibility property: There is a unique group isomorphism from the multiplicative group of the algebraic closure of the finite field with p elements into the group of complex roots of unity whose order is not divisible by p which maps a standard generator of order m to exp(2π i/m). In particular, the minimal polynomials of standard generators of order p^n-1 for all n fulfill the same compatibility conditions as Conway polynomials (see ConwayPolynomial (Reference: ConwayPolynomial)). This can provide an alternative for the lifts used by BrauerCharacterValue (Reference: BrauerCharacterValue) which works for a much wider set of finite field elements where Conway polynomials are very difficult or impossible to compute.

A translation of existing Brauer character tables relative to the lift defined by Conway polynomials to the lift defined by our StandardCyclicGenerator (3.1-1) can be computed with StandardValuesBrauerCharacter (4.7-1), provided the relevant Conway polynomials are known.

The article [Lüb22] also defines a standardized embedding of GAPs finite fields constructed with GF (Reference: GF for field size) into the algebraic closure of the prime field F_p constructed here. This is available with StandardIsomorphismGF (2.4-5).

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