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<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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