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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %A fields.xml GAP documentation Thomas Breuer -->
<!-- %% -->
<!-- %% -->
<!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
<!-- %Y Copyright (C) 2002 The GAP Group -->
<!-- %% -->
<Chapter Label="Fields and Division Rings">
<Heading>Fields and Division Rings</Heading>
<#Include Label="[1]{field}">
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- <Section Label="Field Homomorphisms"> -->
<!-- <Heading>Field Homomorphisms</Heading> -->
<!-- </Section> -->
</Chapter>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %E -->
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