<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Subfields of Fields">
<Heading>Subfields of Fields</Heading>
<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>
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