TableRupt

Computation of Splitting Tables

1 Introduction

This file corresponds to an implementation in Magma of the function for the computation of the Splitting Tables defined in A New Tool For Computing Galois Groups And Galois Ideals[S. Orange, G. Renault, A. Valibouze].

Authors : S. Orange (LMAH, Le Havre University) and G. Renault (LIP6, Paris 6 University).

Source of this file : Documented Magma Code file.

2 Implementation

2.1 SplittingTable

Parameter : d the degree of the transitive groups we want to study
Output : a function which gives all the informations for the splitting tables degree d. This function takes an integer i as input. Its output is defined as follows: if i = −1 the function returns the splitting tables of degree d represented by two sequences, if i = 0 then the function returns the number of lines in the splitting table and if i> 0 the function returns the line number i in the splitting table of degree d. (see the example below).

function SplittingTable(d); Sortie1:=[]; Sortie2:=[]; Sortie1[1]:=[[0,0],[0,0]]; Sortie2[1]:=[0]; for i:=1 to NumberOfTransitiveGroups(d) do Gtest:=TransitiveGroup(d,i); OrbsStab:=Orbits(Stabilizer(Gtest,1)); TypeGroup:=[]; for o in OrbsStab do OrbIm:=OrbitImage(Stabilizer(Gtest,1),o); NumbGroup,DegGroup:=TransitiveGroupIdentification(OrbIm); GroupId:=[DegGroup,NumbGroup]; TypeGroup[#TypeGroup+1]:=GroupId; end for; TypeGroup:=Sort(TypeGroup); Indtest:=Index(Sortie1,TypeGroup); if Indtest eq 0 then Sortie1[#Sortie1+1]:=TypeGroup; Sortie2[#Sortie2+1]:=[i]; else Append(~Sortie2[Indtest],i); end if; end for; Sortie1:=Sortie1[2..#Sortie1]; Sortie2:=Sortie2[2..#Sortie2]; Sortie:= function(ligne) case ligne : when -1 : return Sortie1,Sortie2; when 0 : return #Sortie1; else return Sortie1[ligne],Sortie2[ligne]; end case; end function; return Sortie; end function;

3 Examples

In Magma the function SplittingTable can be used in the following manner:

3.1 Computation of the Splitting Table of degree 12

Here we want to study the splitting table of degree 12.

 

> f12:=SplittingTable(12);
> f12(0);
124
> f12(100);
[
    [ 1, 1 ],
    [ 3, 2 ],
    [ 8, 42 ]
]
[ 251, 254, 276 ]
> f12(119);
[
    [ 1, 1 ],
    [ 1, 1 ],
    [ 10, 39 ]
]
[ 293 ]

In this example one can see that the splitting table of degree 12 has 124 lines. The line number 100 gives the following informations: a polynomial with Galois group in {12T₂₅₁, 12T₂₅₄, 12T₂₇₆} has 3 irreducible factors over its stem field which have Galois group conjugate to 1T₁, 3T₂ and 8T₄₂ respectively. The line number 119 gives the following informations: a polynomial with Galois group conjugates to 12T₂₉₃ has 3 irreducible factors over its stem field which have Galois group equal to 1T₁, 1T₁ and 10T₃₉ respectively.

3.2 Application to constructive inverse Galois problem.

With the informations given by the line 119, one can construct a polynomial with coefficients in a number field with Galois 10T₃₉.

 

> load galpols;                        
Loading "/usr/local/magma/libs/galpols/galpols"
> f:=PolynomialWithGaloisGroup(12,293);
> N<a>:=NumberField(f);                                                    
> PRN<x>:=PolynomialRing(N);           
> Factorization(PRN!f);     
[
    <x - a, 1>,
    <x + a, 1>,
    <x^10 + (a^2 - 13)*x^8 + (a^4 - 13*a^2 + 65)*x^6 + (a^6 - 13*a^4 + 65*a^2 - 
        156)*x^4 + (a^8 - 13*a^6 + 65*a^4 - 156*a^2 + 181)*x^2 + a^10 - 13*a^8 + 
        65*a^6 - 156*a^4 + 181*a^2 - 86, 1>
]

The third polynomial given by Magma after the factorisation has Galois group equals to 10T₃₉.

This document was translated from a Documented Magma Code to L^AT_EX by DMC2TEX beta0.1 version.

This document was translated from L^AT_EX by H^EV^EA.