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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## 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>
## Arguments: base field, polynomial, check
## If check is true, then the irreducibility of the polynomial in
## polynomial ring over base field is checked.
## </Description>
## </ManSection>
##
DeclareOperation( "AlgebraicElementsFamily",
[IsField,IsUnivariatePolynomial,IsBool]);
#############################################################################
##
#O AlgebraicExtension(<K>,<f>)
##
## <#GAPDoc Label="AlgebraicExtension">
## <ManSection>
## <Oper Name="AlgebraicExtension" Arg='K,f[,nam]'/>
## <Oper Name="AlgebraicExtensionNC" Arg='K,f[,nam]'/>
##
## <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 Attr="RootOfDefiningPolynomial"/>
## of <A>L</A> holds a root of <A>f</A> in <A>L</A>.
## By default this root is printed as <C>a</C>, this string can be
## overwritten with the optional argument <A>nam</A>. <P/>
##
## The first version of the command checks that the polynomial <A>f</A>
## is an irreducible polynomial over <A>K</A>. This check is skipped with
## the <C>NC</C> variant.
## <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
## gap> l := AlgebraicExtensionNC(Rationals, x^24+3*x^2+1, "alpha");;
## gap> RootOfDefiningPolynomial(l)^50;
## 9*alpha^6+6*alpha^4+alpha^2
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AlgebraicExtension",
[IsField,IsUnivariatePolynomial]);
DeclareOperation( "AlgebraicExtensionNC",
[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 AlgExtEmbeddedPol(<ext>,<pol>)
##
## <ManSection>
## <Func Name="AlgExtEmbeddedPol" Arg='ext,pol'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("AlgExtEmbeddedPol");
DeclareGlobalFunction("AlgExtSquareHensel");
#############################################################################
##
#F IdealDecompositionsOfPolynomial( <f> [:"onlyone"] ) finds ideal decompositions of rational f
##
## <#GAPDoc Label="IdealDecompositionsOfPolynomial">
## <ManSection>
## <Func Name="IdealDecompositionsOfPolynomial" Arg='pol'/>
##
## <Description>
## Let <M>f</M> be a univariate, rational, irreducible, polynomial. A
## pair <M>g</M>,<M>h</M> of polynomials of degree strictly
## smaller than that of <M>f</M>, such that <M>f(x)|g(h(x))</M> is
## called an ideal decomposition. In the context of field
## extensions, if <M>\alpha</M> is a root of <M>f</M> in a suitable extension
## and <M>Q</M> the field of rational numbers. Such decompositions correspond
## to (proper) subfields <M>Q < Q(\beta) < Q(\alpha)</M>,
## where <M>g</M> is the minimal polynomial of <M>\beta</M>.
## This function determines such decompositions up to equality of the subfields
## <M>Q(\beta)</M>, thus determining subfields of a given algebraic extension.
## It returns a list of pairs <M>[g,h]</M> (and an empty list if no such
## decomposition exists). If the option <A>onlyone</A> is given it returns at
## most one such decomposition (and performs faster).
## <Example><![CDATA[
## gap> x:=X(Rationals,"x");;pol:=x^8-24*x^6+144*x^4-288*x^2+144;;
## gap> l:=IdealDecompositionsOfPolynomial(pol);
## [ [ x^2+72*x+144, x^6-20*x^4+60*x^2-36 ],
## [ x^2-48*x+144, x^6-21*x^4+84*x^2-48 ],
## [ x^2+288*x+17280, x^6-24*x^4+132*x^2-288 ],
## [ x^4-24*x^3+144*x^2-288*x+144, x^2 ] ]
## gap> List(l,x->Value(x[1],x[2])/pol);
## [ x^4-16*x^2-8, x^4-18*x^2+33, x^4-24*x^2+120, 1 ]
## gap> IdealDecompositionsOfPolynomial(pol:onlyone);
## [ [ x^2+72*x+144, x^6-20*x^4+60*x^2-36 ] ]
## ]]></Example>
## In this example the given polynomial is regular with Galois group
## <M>Q_8</M>, as expected we get four proper subfields.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IdealDecompositionsOfPolynomial");
DeclareSynonym("DecomPoly",IdealDecompositionsOfPolynomial);
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