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<Chapter Label="Finite Fields">
<Heading>Finite Fields</Heading>

This chapter describes the special functionality which exists in &GAP; for
finite fields and their elements.
Of course the general functionality for fields
(see Chapter <Ref Chap="Fields and Division Rings"/>) also applies
to finite fields.
<P/>
In the following, the term <E>finite field element</E> is used to denote
&GAP; objects in the category <Ref Filt="IsFFE"/>,
and <E>finite field</E> means a field consisting of such elements.
Note that in principle we must distinguish these fields from (abstract)
finite fields.
For example, the image of the embedding of a finite field into a field of
rational functions in the same characteristic is of course a finite field
but its elements are not in <Ref Filt="IsFFE"/>,
and in fact &GAP; does currently not support such fields.
<P/>
Special representations exist for row vectors and matrices over small
finite fields
(see sections <Ref Sect="Row Vectors over Finite Fields"/>
and <Ref Sect="Matrices over Finite Fields"/>).


<Section Label="Finite Field Elements">
<Heading>Finite Field Elements</Heading>

<#Include Label="IsFFE">
<#Include Label="Z">
<#Include Label="IsLexOrderedFFE">

</Section>


<Section Label="Operations for Finite Field Elements">
<Heading>Operations for Finite Field Elements</Heading>

<#Include Label="[2]{ffe}">
<#Include Label="DegreeFFE">
<#Include Label="LogFFE">
<#Include Label="IntFFE">
<#Include Label="IntFFESymm">
<#Include Label="IntVecFFE">
<#Include Label="AsInternalFFE">
<#Include Label="RootFFE">

</Section>


<Section Label="Creating Finite Fields">
<Heading>Creating Finite Fields</Heading>

<#Include Label="DefaultField:ffe">
<#Include Label="GaloisField">
<#Include Label="PrimitiveRoot">

</Section>


<Section Label="Frobenius Automorphisms">
<Heading>Frobenius Automorphisms</Heading>

<#Include Label="FrobeniusAutomorphism">

</Section>


<Section Label="Conway Polynomials">
<Heading>Conway Polynomials</Heading>

<#Include Label="ConwayPolynomial">
<#Include Label="IsCheapConwayPolynomial">
<#Include Label="RandomPrimitivePolynomial">

</Section>


<Section Label="Printing, Viewing and Displaying Finite Field Elements">
<Heading>Printing, Viewing and Displaying Finite Field Elements</Heading>

<ManSection>
<Meth Name="ViewObj" Arg="z" Label="for a ffe"/>
<Meth Name="PrintObj" Arg="z" Label="for a ffe"/>
<Meth Name="Display" Arg="z" Label="for a ffe"/>

<Description>
Internal finite field elements are viewed, printed and displayed (see
section <Ref Sect="View and Print"/> for the distinctions between these
operations) as powers of the primitive root (except for the zero
element, which is displayed as 0 times the primitive root). Thus:
<P/>
<Example><![CDATA[
gap> Z(2);
Z(2)^0
gap> Z(5)+Z(5);
Z(5)^2
gap> Z(256);
Z(2^8)
gap> Zero(Z(125));
0*Z(5)
]]></Example>
<P/>
Note also that each element is displayed as an element of the field it
generates, and that the size of the field is printed as a power
of the characteristic.
<P/>
Elements of larger fields are printed as &GAP; expressions which
represent them as sums of low powers of the primitive root:
<P/>
<Example><![CDATA[
gap> Print( Z(3,20)^100, "\n" );
2*Z(3,20)^2+Z(3,20)^4+Z(3,20)^6+Z(3,20)^7+2*Z(3,20)^9+2*Z(3,20)^10+2*Z\
(3,20)^12+2*Z(3,20)^15+2*Z(3,20)^17+Z(3,20)^18+Z(3,20)^19
gap> Print( Z(3,20)^((3^20-1)/(3^10-1)), "\n" );
Z(3,20)^3+2*Z(3,20)^4+2*Z(3,20)^7+Z(3,20)^8+2*Z(3,20)^10+Z(3,20)^11+2*\
Z(3,20)^12+Z(3,20)^13+Z(3,20)^14+Z(3,20)^15+Z(3,20)^17+Z(3,20)^18+2*Z(\
3,20)^19
gap> Z(3,20)^((3^20-1)/(3^10-1)) = Z(3,10);
true
]]></Example>
<P/>
Note from the second example above, that these elements are not always
written over the smallest possible field before being output.
<P/>
The <Ref Meth="ViewObj" Label="for a ffe"/> and
<Ref Meth="Display" Label="for a ffe"/> methods
for these large finite field elements use a slightly more compact,
but mathematically equivalent representation.
The primitive root is represented by <C>z</C>; its <M>i</M>-th power by
<C>z</C><M>i</M> and <M>k</M> times this power by <M>k</M><C>z</C><M>i</M>.
<P/>
<Example><![CDATA[
gap> Z(5,20)^100;
z2+z4+4z5+2z6+z8+3z9+4z10+3z12+z13+2z14+4z16+3z17+2z18+2z19
]]></Example>
<P/>
This output format is always used for <Ref Meth="Display" Label="for a ffe"/>.
For <Ref Meth="ViewObj" Label="for a ffe"/> it is used only if its length
would not exceed the number of lines specified in the user preference
<C>ViewLength</C> (see <Ref Func="SetUserPreference"/>. Longer output is
replaced by <C><<an element of GF(<A>p</A>, <A>d</A>)>></C>.
<P/>
<Example><![CDATA[
gap> Z(2,409)^100000;
<<an element of GF(2, 409)>>
gap> Display(Z(2,409)^100000);
z2+z3+z4+z5+z6+z7+z8+z10+z11+z13+z17+z19+z20+z29+z32+z34+z35+z37+z40+z\
45+z46+z48+z50+z52+z54+z55+z58+z59+z60+z66+z67+z68+z70+z74+z79+z80+z81\
+z82+z83+z86+z91+z93+z94+z95+z96+z98+z99+z100+z101+z102+z104+z106+z109\
+z110+z112+z114+z115+z118+z119+z123+z126+z127+z135+z138+z140+z142+z143\
+z146+z147+z154+z159+z161+z162+z168+z170+z171+z173+z174+z181+z182+z183\
+z186+z188+z189+z192+z193+z194+z195+z196+z199+z202+z204+z205+z207+z208\
+z209+z211+z212+z213+z214+z215+z216+z218+z219+z220+z222+z223+z229+z232\
+z235+z236+z237+z238+z240+z243+z244+z248+z250+z251+z256+z258+z262+z263\
+z268+z270+z271+z272+z274+z276+z282+z286+z288+z289+z294+z295+z299+z300\
+z301+z302+z303+z304+z305+z306+z307+z308+z309+z310+z312+z314+z315+z316\
+z320+z321+z322+z324+z325+z326+z327+z330+z332+z335+z337+z338+z341+z344\
+z348+z350+z352+z353+z356+z357+z358+z360+z362+z364+z366+z368+z372+z373\
+z374+z375+z378+z379+z380+z381+z383+z384+z386+z387+z390+z395+z401+z402\
+z406+z408
]]></Example>
<P/>
Finally note that elements of large prime fields are stored and
displayed as residue class objects. So
<P/>
<Example><![CDATA[
gap> Z(65537);
ZmodpZObj( 3, 65537 )
]]></Example>
</Description>
</ManSection>

</Section>
</Chapter>

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