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<!-- %A  fields.msk                   GAP documentation              Thomas Breuer -->
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<!-- %A  @(#)<M>Id: fields.msk,v 1.18 2005/04/21 09:38:23 gap Exp </M> -->
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<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
<!-- %Y  Copyright (C) 2002 The GAP Group -->
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<Chapter Label="Fields and Division Rings">
<Heading>Fields and Division Rings</Heading>

<#Include Label="[1]{field}">


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<Section Label="Generating Fields">
<Heading>Generating Fields</Heading>

<#Include Label="IsDivisionRing">
<#Include Label="IsField">
<#Include Label="Field">
<#Include Label="DefaultField">
<#Include Label="DefaultFieldByGenerators">
<#Include Label="GeneratorsOfDivisionRing">
<#Include Label="GeneratorsOfField">
<#Include Label="DivisionRingByGenerators">
<#Include Label="AsDivisionRing">

</Section>


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<Section Label="Subfields of Fields">
<Heading>Subfields of Fields</Heading>

<#Include Label="Subfield">
<#Include Label="FieldOverItselfByGenerators">
<#Include Label="PrimitiveElement">
<#Include Label="PrimeField">
<#Include Label="IsPrimeField">
<#Include Label="DegreeOverPrimeField">
<#Include Label="DefiningPolynomial">
<#Include Label="RootOfDefiningPolynomial">
<#Include Label="FieldExtension">
<#Include Label="Subfields">

</Section>


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<Section Label="Galois Action">
<Heading>Galois Action</Heading>

<#Include Label="[2]{field}">
<P/>
<#Include Label="[3]{field}">

<Index Key="IsFieldControlledByGaloisGroup"><C>IsFieldControlledByGaloisGroup</C></Index>
<#Include Label="GaloisGroup:field">

<ManSection>
<Oper Name="MinimalPolynomial" Arg='F, z[, ind]' Label="over a field"/>

<Description>
returns the minimal polynomial of <A>z</A> over the field <A>F</A>.
This is a generator of the ideal in <M><A>F</A>[x]</M> of all polynomials
which vanish on <A>z</A>.
(This definition is consistent with the general definition of
<Ref Oper="MinimalPolynomial"/> for rings.)
<P/>
<Example><![CDATA[
gap> MinimalPolynomial( Rationals, E(8) );
x_1^4+1
gap> MinimalPolynomial( CF(4), E(8) );
x_1^2+(-E(4))
gap> MinimalPolynomial( CF(8), E(8) );
x_1+(-E(8))
]]></Example>
</Description>
</ManSection>


<#Include Label="TracePolynomial">
<#Include Label="Norm">
<#Include Label="Trace">
<#Include Label="Conjugates">
<#Include Label="NormalBase">

</Section>


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<!-- <Section Label="Field Homomorphisms"> -->
<!-- <Heading>Field Homomorphisms</Heading> -->
<!-- </Section> -->

</Chapter>


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