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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 declarations for the computation of Galois Groups.
##
#############################################################################
##
#V InfoGalois
##
## <ManSection>
## <InfoClass Name="InfoGalois"/>
##
## <Description>
## is the info class for the Galois group recognition functions.
## </Description>
## </ManSection>
##
DeclareInfoClass("InfoGalois");
#############################################################################
##
#F GaloisType(<f>[,<cand>])
##
## <#GAPDoc Label="GaloisType">
## <ManSection>
## <Attr Name="GaloisType" Arg='f'/>
##
## <Description>
## Let <A>f</A> be an irreducible polynomial with rational coefficients. This
## function returns the type of Gal(<A>f</A>)
## (considered as a transitive permutation group of the roots of <A>f</A>). It
## returns a number <A>i</A> if Gal(<A>f</A>) is permutation isomorphic to
## <C>TransitiveGroup(<A>n</A>,<A>i</A>)</C> where <A>n</A> is the degree of <A>f</A>.
## <P/>
## Identification is performed by factoring
## appropriate Galois resolvents as proposed in <Cite Key="MS85"/>. This function
## is provided for rational polynomials of degree up to 15. However, in some
## cases the required calculations become unfeasibly large.
## <P/>
## For a few polynomials of degree 14, a complete discrimination is not yet
## possible, as it would require computations, that are not feasible with
## current factoring methods.
## <P/>
## This function requires the transitive groups library to be installed (see
## <Ref BookName="transgrp" Sect="Transitive Permutation Groups"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("GaloisType",IsRationalFunction);
#############################################################################
##
#F ProbabilityShapes(<f>)
##
## <#GAPDoc Label="ProbabilityShapes">
## <ManSection>
## <Func Name="ProbabilityShapes" Arg='f'/>
##
## <Description>
## Let <A>f</A> be an irreducible polynomial with rational coefficients. This
## function returns a list of the most likely type(s) of Gal(<A>f</A>)
## (see <Ref Attr="GaloisType"/>), based
## on factorization modulo a set of primes.
## It is very fast, but the result is only probabilistic.
## <P/>
## This function requires the transitive groups library to be installed (see
## <Ref BookName="transgrp" Sect="Transitive Permutation Groups"/>).
## <Example><![CDATA[
## gap> f:=x^9-9*x^7+27*x^5-39*x^3+36*x-8;;
## gap> GaloisType(f);
## 25
## gap> TransitiveGroup(9,25);
## [1/2.S(3)^3]3
## gap> ProbabilityShapes(f);
## [ 25 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("ProbabilityShapes");
DeclareGlobalFunction("SumRootsPol");
DeclareGlobalFunction("ProductRootsPol");
DeclareGlobalFunction("Tschirnhausen");
DeclareGlobalFunction("TwoSeqPol");
DeclareGlobalFunction("GaloisSetResolvent");
DeclareGlobalFunction("GaloisDiffResolvent");
DeclareGlobalFunction("ParityPol");
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