<Chapter Label="ch:Intro">
<Heading>Introduction to &StdFF; package</Heading>
<Section Label="sec:Overview">
<Heading>Aim</Heading>
This &GAP;-package provides a reference implementation for the
standardized constructions of finite fields and generators of cyclic
subgroups defined in the article <Cite Key="StdFFCyc"/>.
<P/>
The main functions are <Ref Func="FF"/> to construct the finite field of
order <M>p^n</M> and <Ref Oper="StandardCyclicGenerator"/> to construct a
standardized generator of the multiplicative subgroup of a given order
<M>m</M> in such a finite field. The condition on <M>m</M> is that it
divides <M>p^n-1</M> and that &GAP; can factorize this number. (The
factorization of the multiplicative group order <M>p^n-1</M> is not
needed.)
<P/>
Each field of order <M>p^n</M> comes with a natural
<M>\mathbb{F}_p</M>-basis which is a subset of the natural basis of each
extension field of order <M>p^{nm}</M>. The union of these bases is a
basis of the algebraic closure of <M>\mathbb{F}_p</M>. Each element
of the algebraic closure can be identified by its degree <M>d</M>
over its prime field and a number <M>0 \leq k \leq p^d-1</M> (see
<Ref Oper="SteinitzPair"/>) or, equivalently, by a certain multivariate
polynomial (see <Ref Meth="AsPolynomial" Label="for elements in standard
finite fields"/>). This can be useful for transferring finite field
elements between programs which use the same construction of finite
fields.
<P/>
The standardized generators of multiplicative cyclic groups have a
nice compatibility property: There is a unique group isomorphism
from the multiplicative group <Alt Not="Text"><Alt Not="HTML
noMathJax">\bar{\mathbb{F}}_p^\times of the algebraic
closure of the finite field with <M>p</M> elements into the group
of complex roots of unity whose order is not divisible by <M>p</M>
which maps a standard generator of order <M>m</M> to <M>\exp(2\pi
i/m)</M>. In particular, the minimal polynomials of standard generators of
order <M>p^n-1</M> for all <M>n</M> fulfill the same compatibility
conditions as Conway polynomials (see <Ref BookName="Reference"
Func="ConwayPolynomial"/>). This can provide an alternative for the lifts
used by <Ref BookName="Reference" Attr="BrauerCharacterValue"/> which
works for a much wider set of finite field elements where Conway
polynomials are very difficult or impossible to compute.
<P/>
A translation of existing Brauer character tables relative to the
lift defined by Conway polynomials to the lift defined by our
<Ref Oper="StandardCyclicGenerator"/> can be computed with <Ref
Func="StandardValuesBrauerCharacter"/>, provided the relevant Conway
polynomials are known.
<P/>
The article <Cite Key="StdFFCyc"/> also defines a standardized embedding
of &GAP;s finite fields constructed with <Ref BookName="Reference"
Func="GF"/> into the algebraic closure of the prime field
<M>\mathbb{F}_p</M> constructed here. This is available with <Ref
Func="StandardIsomorphismGF"/>.
</Section>
</Chapter>
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