/usr/share/gap/lib/algfld.gd is in gap-libs 4r7p5-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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##
#W algfld.gd GAP Library Alexander Hulpke
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1999 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the categories, attributes, properties and operations
## for algebraic extensions of fields and their elements
#############################################################################
##
#C IsAlgebraicElement(<obj>)
##
## <#GAPDoc Label="IsAlgebraicElement">
## <ManSection>
## <Filt Name="IsAlgebraicElement" Arg='obj' Type='Category'/>
##
## <Description>
## is the category for elements of an algebraic extension.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsAlgebraicElement", IsScalar and IsZDFRE and
IsAssociativeElement and IsAdditivelyCommutativeElement
and IsCommutativeElement);
DeclareCategoryCollections( "IsAlgebraicElement");
DeclareCategoryCollections( "IsAlgebraicElementCollection");
DeclareCategoryCollections( "IsAlgebraicElementCollColl");
#############################################################################
##
#C IsAlgebraicElementFamily Category for Families of Algebraic Elements
##
## <ManSection>
## <Filt Name="IsAlgebraicElementFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsAlgebraicElement" );
#############################################################################
##
#C IsAlgebraicExtension(<obj>)
##
## <#GAPDoc Label="IsAlgebraicExtension">
## <ManSection>
## <Filt Name="IsAlgebraicExtension" Arg='obj' Type='Category'/>
##
## <Description>
## is the category of algebraic extensions of fields.
## <Example><![CDATA[
## gap> IsAlgebraicExtension(e);
## true
## gap> IsAlgebraicExtension(Rationals);
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsAlgebraicExtension", IsField );
#############################################################################
##
#A AlgebraicElementsFamilies List of AlgElm. families to one poly over
##
## <ManSection>
## <Attr Name="AlgebraicElementsFamilies" Arg='obj'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "AlgebraicElementsFamilies",
IsUnivariatePolynomial, "mutable" );
#############################################################################
##
#O AlgebraicElementsFamily Create Family of alg elms
##
## <ManSection>
## <Oper Name="AlgebraicElementsFamily" Arg='obj'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareOperation( "AlgebraicElementsFamily",
[IsField,IsUnivariatePolynomial]);
#############################################################################
##
#O AlgebraicExtension(<K>,<f>)
##
## <#GAPDoc Label="AlgebraicExtension">
## <ManSection>
## <Oper Name="AlgebraicExtension" Arg='K,f'/>
##
## <Description>
## constructs an extension <A>L</A> of the field <A>K</A> by one root of the
## irreducible polynomial <A>f</A>, using Kronecker's construction.
## <A>L</A> is a field whose <Ref Attr="LeftActingDomain"/> value is
## <A>K</A>.
## The polynomial <A>f</A> is the <Ref Attr="DefiningPolynomial"/> value
## of <A>L</A> and the attribute
## <Ref Func="RootOfDefiningPolynomial"/>
## of <A>L</A> holds a root of <A>f</A> in <A>L</A>.
## <Example><![CDATA[
## gap> x:=Indeterminate(Rationals,"x");;
## gap> p:=x^4+3*x^2+1;;
## gap> e:=AlgebraicExtension(Rationals,p);
## <algebraic extension over the Rationals of degree 4>
## gap> IsField(e);
## true
## gap> a:=RootOfDefiningPolynomial(e);
## a
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AlgebraicExtension",
[IsField,IsUnivariatePolynomial]);
#############################################################################
##
#F MaxNumeratorCoeffAlgElm(<a>)
##
## <ManSection>
## <Func Name="MaxNumeratorCoeffAlgElm" Arg='a'/>
##
## <Description>
## maximal (absolute value, in numerator)
## coefficient in the representation of algebraic elm. <A>a</A>
## </Description>
## </ManSection>
##
DeclareOperation("MaxNumeratorCoeffAlgElm",[IsScalar]);
#############################################################################
##
#F DefectApproximation( <K> ) . . . . . . . approximation for defect K, i.e.
#F denominators of integer elements in K
##
## <ManSection>
## <Func Name="DefectApproximation" Arg='K'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute("DefectApproximation",IsAlgebraicExtension);
#############################################################################
##
#F AlgExtEmbeddedPol(<ext>,<pol>)
##
## <ManSection>
## <Func Name="AlgExtEmbeddedPol" Arg='ext,pol'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("AlgExtEmbeddedPol");
DeclareGlobalFunction("AlgExtSquareHensel");
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