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<Chapter Label="ch:StdFF">
<Heading>Standard finite fields</Heading>

<Section Label="sec:StdFFIntro">
<Heading>Definition of standard finite fields</Heading>

In <Cite Key="StdFFCyc" /> we define  for each prime <M>p</M> and positive
integer <M>n</M> a standardized model for the finite field with <M>p^n</M>
elements. This is done by defining  for each prime <M>r</M> polynomials of
degree <M>r</M> which define  recursively <M>r</M>-power extensions of the
prime field <M>GF(p)</M> and by combining these  for all <M>r | n</M> in a
unique tower of  extensions of finite fields where  the successive degrees
are non-decreasing primes.
<P/>
Relative  to this  tower of  prime degree  extensions the  resulting field
comes with a natural basis over the prime field which we call the <E>tower
basis</E>. This  construction has the  nice property that whenever  <M>n |
m</M> then  the tower  basis of  the field with  <M>p^n</M> elements  is a
subset of  the tower  basis of  the field  with <M>p^m</M>  elements. (See
<Cite Key="StdFFCyc" /> for more details.)
<P/>
Expressing elements as linear combination of the tower basis we define
a bijection from the elements in the field of order <M>p^n</M> to the
range <C>[0..p^n-1]</C>; we call the number assigned to an element its
<E>Steinitz number</E>.
<P/>
Via  this   construction  each  element   in  the  algebraic   closure  of
<M>GF(p)</M> can be identified by its degree <M>d</M> over the prime field
and its  Steinitz number in  the field  with <M>p^d</M> elements  (we call
this a <E>Steinitz pair</E>).
<P/>
Since arithmetic  in simple  algebraic extensions  is more  efficient than
in  iterated extensions  we  construct the  fields  recursively as  simple
extensions, and  including information about  the base change  between the
tower basis and the basis given by the powers of the generator.
</Section>

<Section Label="sec:CreateFF">
<Heading>Creating standard finite fields</Heading>

<#Include Label="FF">
<#Include Label="IsFF">

</Section>

<Section Label="sec:ElementsFF">
<Heading>Elements in standard finite fields</Heading>

For fields in <Ref  Filt="IsStandardFiniteField"/> we provide functions to
map elements  to their  linear combination  of the  tower basis,  to their
Steinitz number and  Steinitz pair, or to  their representing multivariate
polynomial with respect to all prime degree extensions, and vice versa.
<P/>

<#Include Label="FFElmConv">

</Section>

<Section Label="sec:FFEmbed">
<Heading>Embeddings of standard finite fields</Heading>

The tower  basis of a  standard finite  field <C>F</C> contains  the tower
basis of any subfield. This  yields a construction of canonical embeddings
of  all subfields  of  <C>F</C> into  <C>F</C>. And  one  can easily  read
off  the smallest  subfield containing  an  element in  <C>F</C> from  its
coefficient vector  with respect to the  tower basis. Each element  of the
algebraic closure of  <C>FF(p,1)</C> is uniquely determined  by its degree
<C>d</C> and its Steinitz number in <C>FF(p, d)</C>.

<#Include Label="SteinitzPair">
<#Include Label="FFEmbedding">
<#Include Label="FFArith">
<#Include Label="StandardIsomorphismGF">

</Section>

</Chapter>

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