Fields and Division Rings
<#Include Label="[1]{field}">

Generating Fields
<#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">

Subfields of Fields
<#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">

Galois Action
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<#Include Label="[3]{field}">

IsFieldControlledByGaloisGroup
<#Include Label="GaloisGroup:field">

returns the minimal polynomial of z over the field F. This is a generator of the ideal in F[x] of all polynomials which vanish on z. (This definition is consistent with the general definition of for rings.)

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