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Source Code
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#include <iostream>
#include <string>
#include <vector>
#include <cmath>
#include "lesFonctions.h"
using namespace std;
int main()
{
string choix, cofp, cofq, cofr, cofs, degchoix, reponse;
int iDegChoix;
int disc1(0), disc2(0);
bool reductible = false;
int p(0), q(0), r(0), s(0), g, f;
int pR(0), qR(0), rR(0), racine(0);
int i = 0, t, y = 2;
int car;
bool alter = false;
char rep;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| *Ce programme détermine les groupes de Galois associés aux polynômes de degré 3 ou 4* |" << endl;
cout << "| - Etape 1 il faut choisir le degré du polynome tapez (3) pour le degré 3 tapez (4) pour le degré 4 |" << endl;
cout << "| - Etape 2 il faut choisir la caractéristique du corps 0 pour Q un nombre premier > 3 pour un corps fini |" << endl;
cout << "| - Etape 3 il faut choisir les coefficients; Notons que ces coefficients doivent étre dans Z |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Choisir le degré du polynôme |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout<<"-> ";
cin >> degchoix;
while (1)
{
int a = (int)degchoix[0];
if (degchoix.length() >= 2)
{
cout << "Valeur inccorecte tapez 3 ou 4 : "<<endl;
cout<<"-> ";
cin >> degchoix;
}
else if (a == 51 || a == 52)
{
iDegChoix = a - 48;
break;
}
else
{
cout << "Valeur inccorecte tapez 3 ou 4 : "<<endl;
cout <<"-> ";
cin >> degchoix;
}
}
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Choisir la caractéristique du corps |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout << "-> ";
cin >> choix;
int leChoix = verChoix(choix);
do
{
if (leChoix == 1)
{
cout << "veuillez saisir un autre nombre, soit 0 soit un nombre premier > 3 : " << endl;
cout <<"-> ";
cin >> choix;
leChoix = verChoix(choix);
}
else
{
car = leChoix;
break;
}
} while (1);
if (iDegChoix == 3)
{
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << " Polynome de degré 3 p(X) = X³ + pX² + qX + r sur un corps de caractéristique " << car << " " << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout << "Donnez p, q, r dans l'ordre :" << endl;
cout << "-> p = ";
cin >> cofp;
cout << "-> q = ";
cin >> cofq;
cout << "-> r = ";
cin >> cofr;
do
{
if (verCof(cofp, p) && verCof(cofq, q) && verCof(cofr, r))
{
break;
}
else
{
cout << "Valeurs inccorects veuillez introduire d'autres valeurs :" << endl;
cout << "-> p = ";
cin >> cofp;
cout << "-> q = ";
cin >> cofq;
cout << "-> r = ";
cin >> cofr;
p = 0;
q = 0;
r = 0;
continue;
}
} while (1);
if (car != 0)
{
p = ((p % car) + car) % car;
q = ((q % car) + car) % car;
r = ((r % car) + car) % car;
}
cout << endl;
cout << "----> p(X) = X³ + "
<< " (" << p << ")"
<< "X² + "
<< " (" << q << ")"
<< "X + " << r << " **" << endl;
cout << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Test d'irréductibilité |" << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
do
{
if (polyRed(p, q, r, car, iDegChoix))
{
if (car != 0)
{
p = ((p % car) + car) % car;
q = ((q % car) + car) % car;
r = ((r % car) + car) % car;
}
cout << "----> p(X) = X³ + "
<< " (" << p << ")"
<< "X² + "
<< " (" << q << ")"
<< "X + " << r << " est irréductible " << endl;
break;
}
else
{
continue;
}
} while (1);
cout << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Calcule du discriminant (∆) |" << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
long int det(0);
if (reductible == false)
{
det = calDet(p, q, r, car);
cout << endl;
cout << "----> ∆ de "
<< " X³ + "
<< " (" << p << ")"
<< "X² + "
<< " (" << q << ")"
<< "X + " << r << " = " << det << endl;
if (carre(det, car) && car == 0)
{
cout << "----> ∆ est un carré (" << sqrt(det) << ")² = " << det << endl;
cout << endl;
}
else if (carre(det, car) == false && car == 0)
{
cout << "----> ∆ n'est pas un carré" << endl;
cout << endl;
}
else if (carre(det, car) && car != 0)
{
g = 0;
f;
while (g < car)
{
f = pow(g, 2);
if (f % car == det)
{
break;
}
g++;
}
cout << "----> ∆ est un carré " << g << "² ≡ " << det << "[" << car<<"]" << endl;
cout << endl;
}
else if (carre(det, car) == false && car != 0)
{
cout << "----> ∆ n'est pas un carré" << endl;
cout << endl;
}
cout << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Detérmination du groupe de Galois associé |" << endl;
cout << "-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
alter = carre(det, car);
if (alter)
{
cout << "******************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe altérné d'ordre 3 ( A3 )*" << endl;
cout << "******************************************************************************" << endl;
}
else
{
cout << "*********************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe symétrique d'ordre 3 ( S3 )*" << endl;
cout << "*********************************************************************************" << endl;
}
}
}
else
{
bool redDegQ;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << " Polynome de degré 4 f(X) = X⁴ + pX³ + qX² + rX + s sur un corps de caractéristique " << car << " " << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout << "Donnez p, q, r, s dans l'ordre :" << endl;
cout << "-> p = ";
cin >> cofp;
cout << "-> q = ";
cin >> cofq;
cout << "-> r = ";
cin >> cofr;
cout << "-> s = ";
cin >> cofs;
do
{
if (verCof(cofp, p) && verCof(cofq, q) && verCof(cofr, r) && verCof(cofs, s))
{
break;
}
else
{
cout << "Valeurs inccorects veuillez introduire d'autres valeurs :" << endl;
cout << "-> p = ";
cin >> cofp;
cout << "-> q = ";
cin >> cofq;
cout << "-> r = ";
cin >> cofr;
cout << "-> s = ";
cin >> cofs;
p = 0;
q = 0;
r = 0;
s = 0;
continue;
}
} while (1);
if (car != 0)
{
p = ((p % car) + car) % car;
q = ((q % car) + car) % car;
r = ((r % car) + car) % car;
s = ((s % car) + car) % car;
}
cout << endl;
cout << "----> f(X) = X⁴ + "
<< " (" << p << ")"
<< "X³ + "
<< " (" << q << ")"
<< "X² + "
<< "(" << r << ")X + " << s << endl;
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Résolvente cubique associée φ(X) |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
pR = -q;
qR = (p * r) - (4 * s);
rR = ((pow(p, 2) * s) + (pow(r, 2)) - (4 * q * s)) * (-1);
if (car != 0)
{
pR = ((pR % car) + car) % car;
qR = ((qR % car) + car) % car;
rR = ((rR % car) + car) % car;
}
cout << "----> φ(X) = X³ + "
<< " (" << pR << ")"
<< "X² + "
<< " (" << qR << ")"
<< "X + (" << rR << ")" << endl;
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Test d'irréductibilité de φ(X) |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
if (polyRed(pR, qR, rR, car, iDegChoix))
{
cout << "----> φ(X)= X³ + "
<< " (" << pR << ")"
<< "X² + "
<< " (" << qR << ")"
<< "X + (" << rR << ") est irréductible " << endl;
redDegQ = false;
}
else
{
cout << "----> φ(X) = X³ + "
<< " (" << pR << ")"
<< "X² + "
<< " (" << qR << ")"
<< "X + (" << rR << ") est réductible " << endl;
redDegQ = true;
}
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Calcule du discriminant (∆) de φ(X) |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
long int det(0);
if (reductible == false)
{
det = calDet(pR, qR, rR, car);
cout << endl;
cout << "----> ∆ de "
<< " X³ + "
<< " (" << pR << ")"
<< "X² + "
<< " (" << qR << ")"
<< "X + (" << rR << ") = " << det
<< endl;
cout << endl;
if (carre(det, car) && car == 0)
{
cout << "----> ∆ est un carré (" << sqrt(det) << ")² = " << det << endl;
cout << endl;
}
else if (carre(det, car) == false && car == 0)
{
cout << "----> ∆ n'est pas un carré" << endl;
cout << endl;
}
else if (carre(det, car) && car != 0)
{
g = 0;
f;
while (g < car)
{
f = pow(g, 2);
if (f % car == det)
{
break;
}
g++;
}
cout << "----> ∆ est un carré " << g << "² ≡ " << det << "[" << car<<"]" << endl;
cout << endl;
}
else if (carre(det, car) == false && car != 0)
{
cout << "----> ∆ n'est pas un carré" << endl;
cout << endl;
}
}
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Detérmination du groupe de Galois associé |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
if (carre(det, car))
{
if (redDegQ)
{
cout << "******************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe de Klein d'ordre 4 ( V )*" << endl;
cout << "******************************************************************************" << endl;
}
else
{
cout << "******************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe altérné d'ordre 4 ( A4 )*" << endl;
cout << "******************************************************************************" << endl;
}
}
else
{
if (redDegQ == false)
{
cout << "*********************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe symétrique d'ordre 4 ( S4 )*" << endl;
cout << "*********************************************************************************" << endl;
}
else
{
if (det < 0)
{
cout << "*********************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe Diédral d'ordre 8 ( D4 ) *" << endl;
cout << "*********************************************************************************" << endl;
}
else
{
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Recherche a l'aide de la mhétode de Kappe, Warren |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
int j(0);
bool racineTrouve = false;
if (car == 0)
{
while (racineTrouve == false)
{
if ((pow(j, 3) + pR * (pow(j, 2)) + qR * (j) + rR) == 0)
{
racine = j;
racineTrouve = true;
}
else if ((pow(j * (-1), 3) + pR * (pow(j * (-1), 2)) + qR * (j * (-1)) + rR) == 0)
{
racine = j * (-1);
racineTrouve = true;
}
j++;
}
}
else if (car != 0)
{
while (racineTrouve == false && j < car)
{
int b = (pow(j, 3) + (pR * (pow(j, 2))) + (qR * (j)) + rR);
b = b % car;
if (b == 0)
{
racine = j;
racineTrouve = true;
}
j++;
}
}
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Recherche de la racine (β) de la résolvante cubique sur le corps de base |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
cout << "----> (β) = " << racine<< endl;
cout << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << "| Polynômes associés |" << endl;
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
cout << endl;
disc1 = (pow(p, 2) - 4 * (q - racine)) * det;
disc2 = (pow(racine, 2) - (4 * s)) * det;
if (car != 0)
{
disc1 = ((disc1 % car) + car) % car;
disc2 = ((disc2 % car) + car) % car;
}
float cardisc1 = sqrt(disc1);
float cardisc2 = sqrt(disc2);
if (car == 0 && disc1 > 0 && disc2 > 0)
{
cout << "----> X² + (" << p << ")X + (" << q << ")-(" << racine << "), ∆1 x ∆ = " << disc1 << " = (" << cardisc1 << ")²" << endl;
cout << "----> X² - (" << racine << ")X + (" << s << "), ∆2 x ∆ = " << disc2 << " = (" << cardisc2 << ")²" << endl;
cout << endl;
}
else if (car == 0 && (disc1 < 0 || disc2 < 0))
{
cout << "----> X² + (" << p << ")X + (" << q << ")-(" << racine << "), ∆1 x ∆ = " << disc1 << endl;
cout << "----> X² - (" << racine << ")X + (" << s << "), ∆2 x ∆ = " << disc2 << endl;
cout << endl;
}
else if (car != 0)
{
cout << "----> X² + (" << p % car << ")X + (" << q % car << ")-(" << racine % car << "), ∆1 x ∆ = " << disc1 % car << endl;
cout << "----> X² - (" << racine % car << ")X + (" << s % car << "), ∆2 x ∆ = " << disc2 % car << endl;
cout << endl;
}
cout << "--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl;
bool v1 = carre(disc1, car);
bool v2 = carre(disc2, car);
if (car == 0)
{
if (v1)
{
cout << "----> ∆1 est un carré (" << sqrt(disc1) << ")² = " << disc1 << endl;
cout << endl;
}
else
{
cout << "----> ∆1 n'est pas un carré" << endl;
cout << endl;
}
if (v2)
{
cout << "----> ∆2 est un carré (" << sqrt(disc2) << ")² = " << disc2 << endl;
cout << endl;
}
else
{
cout << "----> ∆2 n'est pas un carré" << endl;
cout << endl;
}
}
else
{
if (v1)
{
g = 1;
f;
while (g < car)
{
f = pow(g, 2);
if (f % car == disc1)
{
break;
}
g++;
}
cout << "----> ∆1 est un carré " << g << "² ≡ " << disc1 << " [" << car<<"]" << endl;
cout << endl;
}
else
{
cout << "----> ∆1 n'est pas un carré " << endl;
cout << endl;
}
if (v2)
{
g = 1;
f;
while (g < car)
{
f = pow(g, 2);
if (f % car == disc2)
{
break;
}
g++;
}
cout << "----> ∆2 est un carré " << g << "² ≡ " << disc2 << "[" << car<<"]" << endl;
cout << endl;
}
else
{
cout << "----> ∆2 n'est pas un carré " << endl;
cout << endl;
}
}
if (v1 == true && v2 == true)
{
cout << "*********************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe additif d'ordre 4 ( Z/4Z ) *" << endl;
cout << "*********************************************************************************" << endl;
}
else if (v1 == false || v2 == false)
{
cout << "*********************************************************************************" << endl;
cout << "*Le groupe de Galois associé est isomorphe au groupe Diédral d'ordre 8 ( D4 ) *" << endl;
cout << "*********************************************************************************" << endl;
}
}
}
}
}
return 0;
}
#include "lesFonctions.h"
#include <string>
#include <iostream>
#include <cmath>
int verChoix(std::string choix)
{
bool nombrePremier = true;
bool toutEstOk = true;
int car = 0, i = 0, t = 0, y;
while (i < choix.length())
{
t = (int)choix[i];
if (t > 47 && t < 58)
{
t = t - 48;
car = car + t;
}
else
{
toutEstOk = false;
break;
}
if (i != choix.length() - 1)
{
car = car * 10;
}
i++;
}
y = 2;
while (y < car && nombrePremier == true)
{
if (car % y == 0)
{
nombrePremier = false;
}
else
{
nombrePremier = true;
}
y++;
}
if (nombrePremier && toutEstOk && car != 2 && car != 3)
{
return car;
}
else
{
return 1;
}
}
bool verCof(std::string cof, int &cofe)
{
bool cofOk = true, valeurNegatif = false;
int i = 0, t = 0;
while (i < cof.length())
{
t = (int)cof[i];
if (i == 0 && t == 45)
{
valeurNegatif = true;
i++;
}
t = (int)cof[i];
if (t > 47 && t < 58)
{
t = t - 48;
cofe = cofe + t;
}
else
{
cofOk = false;
break;
}
if (i != cof.length() - 1)
{
cofe = cofe * 10;
}
i++;
}
if (cofOk)
{
if (valeurNegatif)
{
cofe = cofe * (-1);
}
return true;
}
else
{
return false;
}
}
bool polyRed(int &p, int &q, int &r, int car , int iDegchoix)
{
int i = 1;
std::string cofp, cofq, cofr;
bool reductible = false;
if (car == 0 && r != 0)
{
while (i <= abs(r) && reductible == false)
{
if (abs(r) % i == 0)
{
if ((pow(i, 3) + p * (pow(i, 2)) + q * (i) + r) == 0 || (pow(i * (-1), 3) + p * (pow(i * (-1), 2)) + q * (i * (-1)) + r) == 0)
{
reductible = true;
}
}
i++;
}
}
else if (car != 0 && r != 0)
{
i = 0;
while (i < car && reductible == false)
{
int b = (pow(i, 3) + p * (pow(i, 2)) + q * (i) + r);
b = b % car;
if (b == 0)
{
reductible = true;
}
i++;
}
}
if (r == 0)
{
reductible = true;
}
if (reductible && iDegchoix == 3)
{
std::cout << "le polynome est reductible, veuillez saisir d'autres valeurs : " << std::endl;
do
{
std::cout << "-> p = ";
std::cin >> cofp;
std::cout << "-> q = ";
std::cin >> cofq;
std::cout << "-> r = ";
std::cin >> cofr;
p = 0;
q = 0;
r = 0;
if (verCof(cofp, p) && verCof(cofq, q) && verCof(cofr, r))
{
break;
}
else
{
continue;
}
} while (1);
}
if (reductible == true)
{
return false;
}
else
{
return true;
}
}
int calDet(int p, int q, int r, int car)
{
long long int det;
det = (18 * p * q * r) - (4 * (pow(p, 3)) * r) + (pow(p, 2) * pow(q, 2)) - (4 * (pow(q, 3))) - (27 * (pow(r, 2)));
if (car != 0)
{
if (det < 0)
{
det = ((det % car) + car) % car;
}
else
{
det = det % car;
}
}
return det;
}
bool carre(int dete, int car)
{
int i ;
bool eCarre = true ;
if (dete < 0 && car == 0)
{
eCarre = false;
}
else if (dete > 0 && car == 0)
{
std::string disc = std::to_string(sqrt(dete));
i = 0;
while (i < disc.length())
{
if (char l = (int)disc[i] == 46)
{
break;
}
else{
i++;
}
}
int j = i + 1 ;
while (j < disc.length() && eCarre == true)
{
if (disc[j] == 48)
{
eCarre = true ;
j++;
}
else{
eCarre = false ;
}
}
}
if (car != 0)
{
i = 0;
int j;
while (i < car )
{
j = pow(i,2);
if (j % car == dete)
{
eCarre = true;
break;
}
else
{
eCarre = false;
}
i++;
}
}
return eCarre ;
}
#ifndef LESFONCTIONS_H_INCLUDED
#define LESFONCTIONS_H_INCLUDED
#include <string>
#include<iostream>
int verChoix(std::string choix );
bool verCof(std::string cof, int& cofe);
bool polyRed(int& p , int& q , int& r , int car , int iDegchoix);
int calDet(int p , int q , int r , int car) ;
bool carre(int dete , int car);
#endif
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