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| -rw-r--r-- | divers/sageries/equation-de-degre-5 | 58 |
1 files changed, 58 insertions, 0 deletions
diff --git a/divers/sageries/equation-de-degre-5 b/divers/sageries/equation-de-degre-5 new file mode 100644 index 0000000..d542593 --- /dev/null +++ b/divers/sageries/equation-de-degre-5 @@ -0,0 +1,58 @@ +R.<t> = QQ['t'] +f = t^5-5*t+12 +F.<zeta> = CyclotomicField(5) +L.<u> = F.extension(t^2-10*t+1025) +L0.<x0> = L.absolute_field() +(from_L0, to_L0) = L0.structure() +K.<v> = L0.extension(f) +K0.<y0> = K.absolute_field() +(from_K0, to_K0) = K0.structure() +tmproots = [x for (x,_) in f.change_ring(K0).roots()] +g = R(prod([t-(tmproots[i]+tmproots[j]) for i in range(5) for j in range(i)])) +# g.factor() +g1 = t^5 - 5*t^3 - 10*t^2 + 30*t - 36 +reorder = [[i for i in range(5) if tmproots[i] == to_K0(v)][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +if sum([tmproots[reorder[i]]*tmproots[reorder[(i+1)%5]]^3 for i in range(5)]) != to_K0(to_L0(u)): + reorder = [reorder[i] for i in [0,4,3,2,1]] + +roots = [tmproots[i] for i in reorder] +tmpzeta1 = to_K0(to_L0(zeta)) +tmpzeta = [tmpzeta1^i for i in range(5)] +alpha = [sum(roots[i]*tmpzeta[(i*j)%5] for i in range(5)) for j in range(5)] +alpha5 = [from_K0(alpha[i]^5).vector()[0] for i in range(5)] +sqrt5 = 1+2*(zeta+zeta^-1) +sqrtm10m2sqrt5 = 2*(zeta-zeta^-1) +sqrtm10p2sqrt5 = 2*(zeta^2-zeta^-2) +sqrtm10 = -1/2 + u/10 +sqrtm2 = sqrt5*sqrtm10/5 +sqrt25p5sqrt5 = -sqrtm10 * sqrtm10m2sqrt5 / 2 +sqrt25m5sqrt5 = -sqrtm10 * sqrtm10p2sqrt5 / 2 +nice_basis = [to_L0(x) for x in [1, sqrt5, sqrtm10m2sqrt5, sqrtm10p2sqrt5, sqrtm10, sqrtm2, sqrt25p5sqrt5, sqrt25m5sqrt5]] +m = Matrix(QQ, 8, 8, [(QQ^8)((x0.coordinates_in_terms_of_powers())(x)) for x in nice_basis]) +minv = m.inverse() +alpha5_on_x0_basis = [(QQ^8)((x0.coordinates_in_terms_of_powers())(x)) for x in alpha5] +alpha5_on_nice_basis = [v * m.inverse() for v in alpha5_on_x0_basis] + +tmproots = [x for (x,_) in f.change_ring(QQbar).roots()] +reorder = [0] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +reorder += [[i for i in range(5) if g1(tmproots[reorder[len(reorder)-1]] + tmproots[i]) == 0 and not i in reorder][0]] +if sum([tmproots[reorder[i]]*tmproots[reorder[(i+1)%5]]^3 for i in range(5)]) != QQbar(5+10*sqrt(-10)): + reorder = [reorder[i] for i in [0,4,3,2,1]] + +rootsQQbar = [tmproots[i] for i in reorder] +symbolic_basis = [1, sqrt(5), sqrt(-10-2*sqrt(5)), sqrt(-10+2*sqrt(5)), sqrt(-10), sqrt(-2), sqrt(25+5*sqrt(5)), sqrt(25-5*sqrt(5))] +symbolic_alpha5 = [sum([alpha5_on_nice_basis[i][j]*symbolic_basis[j] for j in range(8)]) for i in range(5)] +zetaQQbar = QQbar(exp(2*I*pi/5)) +alphaQQbar = [sum(rootsQQbar[i]*zetaQQbar^((i*j)%5) for i in range(5)) for j in range(5)] +zetab = [0]+[ZZ(floor(arg(CC(alphaQQbar[i]/QQbar(symbolic_alpha5[i]^(1/5))))/N(2*pi/5)+0.5)) for i in range(1,5)] + +symbolic_zetapow = [1, (-symbolic_basis[0]+symbolic_basis[1]+symbolic_basis[2])/4, (-symbolic_basis[0]-symbolic_basis[1]+symbolic_basis[3])/4, (-symbolic_basis[0]-symbolic_basis[1]-symbolic_basis[3])/4, (-symbolic_basis[0]+symbolic_basis[1]-symbolic_basis[2])/4] +symbolic_alpha = [symbolic_zetapow[zetab[i]]*symbolic_alpha5[i]^(1/5) for i in range(5)] +symbolic_roots = [sum([symbolic_zetapow[(zetab[i]+4*i*j)%5]*symbolic_alpha5[i]^(1/5) for i in range(5)])/5 for j in range(5)] |