This GAP-package provides a reference implementation for the standardized constructions of finite fields and generators of cyclic subgroups defined in the article [Lüb23].
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 \(\mathbb{F}_p\)-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 \(\mathbb{F}_p\). Each element of the algebraic closure can be identified by its degree \(d\) over its prime field and a number \(0 \leq k \leq p^d-1\) (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 \(\bar{\mathbb{F}}_p^\times\) 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 \(\exp(2\pi i/m)\). 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 [Lüb23] 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 \(\mathbb{F}_p\) constructed here. This is available with StandardIsomorphismGF (2.4-5).
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