Logging to /Users/s/tmp/pari-12.23 GPRC Done. GP/PARI CALCULATOR Version 2.13.0 (released) i386 running darwin (x86-64/GMP-6.2.0 kernel) 64-bit version compiled: Oct 31 2020, Apple clang version 12.0.0 (clang-1200.0.32.21) threading engine: single (readline v8.0 enabled, extended help enabled) Copyright (C) 2000-2020 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisizemax = 900001792, primelimit = 1000000 (13:19) gp > P=x^3-x-2 %1 = x^3 - x - 2 (13:19) gp > factor(P) %2 = [x^3 - x - 2 1] (13:20) gp > polgalois(P) %3 = [6, -1, 1, "S3"] (13:20) gp > polroots(P) %4 = [1.5213797068045675696040808322544385144 + 0.E-38*I, -0.76068985340228378480204041612721925721 - 0.85787362659517863641803208364326993577*I, -0.76068985340228378480204041612721925721 + 0.85787362659517863641803208364326993577*I]~ (13:20) gp > u=[u1,u2,u3] %5 = [u1, u2, u3] (13:20) gp > X=polroots(P) %6 = [1.5213797068045675696040808322544385144 + 0.E-38*I, -0.76068985340228378480204041612721925721 - 0.85787362659517863641803208364326993577*I, -0.76068985340228378480204041612721925721 + 0.85787362659517863641803208364326993577*I]~ (13:20) gp > X[1] %7 = 1.5213797068045675696040808322544385144 + 0.E-38*I (13:21) gp > X[2] %8 = -0.76068985340228378480204041612721925721 - 0.85787362659517863641803208364326993577*I (13:21) gp > x[3] *** at top-level: x[3] *** ^--- *** incorrect type in _[_] OCcompo1 [not a vector] (t_POL). *** Break loop: type 'break' to go back to GP prompt break> break (13:21) gp > X[3] %9 = -0.76068985340228378480204041612721925721 + 0.85787362659517863641803208364326993577*I (13:21) gp > u1 %10 = u1 (13:21) gp > u[1] %11 = u1 (13:21) gp > ?forperm forperm(a,p,seq): the sequence is evaluated, p going through permutations of a. (13:22) gp > R=1 %12 = 1 (13:22) gp > ??forperm forperm(a,p,seq): Evaluates seq, where the formal variable p goes through some permutations given by a t_VECSMALL. If a is a positive integer then P goes through the permutations of {1, 2, ..., a} in lexicographic order and if a is a small vector then p goes through the (multi)permutations lexicographically larger than or equal to a. ? forperm(3, p, print(p)) Vecsmall([1, 2, 3]) Vecsmall([1, 3, 2]) Vecsmall([2, 1, 3]) Vecsmall([2, 3, 1]) Vecsmall([3, 1, 2]) Vecsmall([3, 2, 1]) When a is itself a t_VECSMALL or a t_VEC then p iterates through multipermutations ? forperm([2,1,1,3], p, print(p)) Vecsmall([2, 1, 1, 3]) Vecsmall([2, 1, 3, 1]) Vecsmall([2, 3, 1, 1]) Vecsmall([3, 1, 1, 2]) Vecsmall([3, 1, 2, 1]) Vecsmall([3, 2, 1, 1]) (13:22) gp > forperm(3,s,print(s)) Vecsmall([1, 2, 3]) Vecsmall([1, 3, 2]) Vecsmall([2, 1, 3]) Vecsmall([2, 3, 1]) Vecsmall([3, 1, 2]) Vecsmall([3, 2, 1]) (13:22) gp > forperm(3,s,R=R*(y-sum(n=1,poldegree(P),X[s[n]]*u[n]))) (13:23) gp > R %15 = y^6 + ((-2.0571151139390031389 E-38 + 0.E-38*I)*u1 + ((-2.938735877055718770 E-39 + 0.E-38*I)*u2 + (0.E-38 + 0.E-38*I)*u3))*y^5 + ((-2.0000000000000000000000000000000000000 + 1.7632415262334312620 E-38*I)*u1^2 + ((2.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2 + (2.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u3)*u1 + ((-2.0000000000000000000000000000000000000 + 0.E-38*I)*u2^2 + (2.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (-2.0000000000000000000000000000000000000 + 0.E-38*I)*u3^2))*y^4 + ((-4.0000000000000000000000000000000000000 + 1.1754943508222875080 E-38*I)*u1^3 + ((6.0000000000000000000000000000000000000 + 0.E-37*I)*u2 + (6.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3)*u1^2 + ((6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u2^2 + (-24.000000000000000000000000000000000000 + 4.114230227878006278 E-38*I)*u3*u2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2)*u1 + ((-4.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2^3 + (6.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3*u2^2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2*u2 + (-4.0000000000000000000000000000000000000 + 0.E-38*I)*u3^3))*y^3 + ((1.0000000000000000000000000000000000000 - 1.7632415262334312620 E-38*I)*u1^4 + ((-1.9999999999999999999999999999999999999 + 1.1754943508222875080 E-38*I)*u2 + (-2.0000000000000000000000000000000000001 + 4.701977403289150032 E-38*I)*u3)*u1^3 + ((2.9999999999999999999999999999999999999 + 0.E-37*I)*u2^2 + (4.701977403289150032 E-38 - 2.350988701644575016 E-38*I)*u3*u2 + (3.0000000000000000000000000000000000001 - 4.701977403289150032 E-38*I)*u3^2)*u1^2 + ((-1.9999999999999999999999999999999999999 - 3.526483052466862524 E-38*I)*u2^3 + (1.880790961315660013 E-37 + 9.403954806578300064 E-38*I)*u3*u2^2 + (-9.403954806578300064 E-38 - 5.289724578700293786 E-38*I)*u3^2*u2 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-37*I)*u3^3)*u1 + ((1.0000000000000000000000000000000000000 - 5.877471754111437540 E-39*I)*u2^4 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-38*I)*u3*u2^3 + (3.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2*u2^2 + (-2.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3*u2 + (1.0000000000000000000000000000000000000 + 0.E-38*I)*u3^4))*y^2 + ((4.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u1^5 + ((-10.000000000000000000000000000000000000 + 9.403954806578300064 E-38*I)*u2 + (-10.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3)*u1^4 + ((3.9999999999999999999999999999999999999 + 9.403954806578300064 E-38*I)*u2^2 + (32.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (4.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2)*u1^3 + ((4.0000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u2^3 + (-24.000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u3*u2^2 + (-24.000000000000000000000000000000000000 - 1.880790961315660013 E-37*I)*u3^2*u2 + (4.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3)*u1^2 + ((-10.000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u2^4 + (32.000000000000000000000000000000000000 + 7.052966104933725048 E-38*I)*u3*u2^3 + (-24.000000000000000000000000000000000000 + 1.6456920911512025112 E-37*I)*u3^2*u2^2 + (32.000000000000000000000000000000000000[+++] (13:24) gp > R=1 %16 = 1 (13:24) gp > X %17 = [1.5213797068045675696040808322544385144 + 0.E-38*I, -0.76068985340228378480204041612721925721 - 0.85787362659517863641803208364326993577*I, -0.76068985340228378480204041612721925721 + 0.85787362659517863641803208364326993577*I]~ (13:24) gp > u %18 = [u1, u2, u3] (13:24) gp > v=[1,2,3] %19 = [1, 2, 3] (13:24) gp > R=1;forperm(3,s,R=R*(y-sum(n=1,poldegree(P),X[s[n]]*u[n]))) (13:24) gp > R %21 = y^6 + ((-2.0571151139390031389 E-38 + 0.E-38*I)*u1 + ((-2.938735877055718770 E-39 + 0.E-38*I)*u2 + (0.E-38 + 0.E-38*I)*u3))*y^5 + ((-2.0000000000000000000000000000000000000 + 1.7632415262334312620 E-38*I)*u1^2 + ((2.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2 + (2.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u3)*u1 + ((-2.0000000000000000000000000000000000000 + 0.E-38*I)*u2^2 + (2.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (-2.0000000000000000000000000000000000000 + 0.E-38*I)*u3^2))*y^4 + ((-4.0000000000000000000000000000000000000 + 1.1754943508222875080 E-38*I)*u1^3 + ((6.0000000000000000000000000000000000000 + 0.E-37*I)*u2 + (6.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3)*u1^2 + ((6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u2^2 + (-24.000000000000000000000000000000000000 + 4.114230227878006278 E-38*I)*u3*u2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2)*u1 + ((-4.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2^3 + (6.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3*u2^2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2*u2 + (-4.0000000000000000000000000000000000000 + 0.E-38*I)*u3^3))*y^3 + ((1.0000000000000000000000000000000000000 - 1.7632415262334312620 E-38*I)*u1^4 + ((-1.9999999999999999999999999999999999999 + 1.1754943508222875080 E-38*I)*u2 + (-2.0000000000000000000000000000000000001 + 4.701977403289150032 E-38*I)*u3)*u1^3 + ((2.9999999999999999999999999999999999999 + 0.E-37*I)*u2^2 + (4.701977403289150032 E-38 - 2.350988701644575016 E-38*I)*u3*u2 + (3.0000000000000000000000000000000000001 - 4.701977403289150032 E-38*I)*u3^2)*u1^2 + ((-1.9999999999999999999999999999999999999 - 3.526483052466862524 E-38*I)*u2^3 + (1.880790961315660013 E-37 + 9.403954806578300064 E-38*I)*u3*u2^2 + (-9.403954806578300064 E-38 - 5.289724578700293786 E-38*I)*u3^2*u2 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-37*I)*u3^3)*u1 + ((1.0000000000000000000000000000000000000 - 5.877471754111437540 E-39*I)*u2^4 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-38*I)*u3*u2^3 + (3.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2*u2^2 + (-2.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3*u2 + (1.0000000000000000000000000000000000000 + 0.E-38*I)*u3^4))*y^2 + ((4.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u1^5 + ((-10.000000000000000000000000000000000000 + 9.403954806578300064 E-38*I)*u2 + (-10.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3)*u1^4 + ((3.9999999999999999999999999999999999999 + 9.403954806578300064 E-38*I)*u2^2 + (32.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (4.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2)*u1^3 + ((4.0000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u2^3 + (-24.000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u3*u2^2 + (-24.000000000000000000000000000000000000 - 1.880790961315660013 E-37*I)*u3^2*u2 + (4.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3)*u1^2 + ((-10.000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u2^4 + (32.000000000000000000000000000000000000 + 7.052966104933725048 E-38*I)*u3*u2^3 + (-24.000000000000000000000000000000000000 + 1.6456920911512025112 E-37*I)*u3^2*u2^2 + (32.000000000000000000000000000000000000 + 9.403954806578300064 E-38*I)*u3^3*u2 + (-10.000000000000000000000000000000000000 - 1.1754943508222875080 E-37*I)*u3^4)*u1 + ((4.0000000000000000000000000000000000000 + 1.1754943508222875080 E-38*I)*u2^5 + (-10.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^4 + (4.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^3 + (4.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3*u2^2 + (-10.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^4*u2 + (4.0000000000000000000000000000000000000 + 0.E-37*I)*u3^5))*y + ((4.0000000000000000000000000000000000000 + 0.E-38*I)*u1^6 + ((-12.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u2 + (-12.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3)*u1^5 + ((23.000000000000000000000000000000000000 + 4.701977403289150032 E-38*I)*u2^2 + (14.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (23.000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2)*u1^4 + ((-26.000000000000000000000000000000000000 + 0.E-37*I)*u2^3 + (-14.000000000000000000000000000000000000 - 1.4105932209867450096 E-37*I)*u3*u2^2 + (-14.000000000000000000000000000000000000 + 4.701977403289150032 E-38*I)*u3^2*u2 + (-26.000000000000000000000000000000000000 + 0.E-37*I)*u3^3)*u1^3 + ((23.000000000000000000000000000000000000 + 0.E-37*I)*u2^4 + (-14.000000000000000000000000000000000000 + 1.4105932209867450096 E-37*I)*u3*u2^3 + (42.00000000000000000[+++] (13:25) gp > substvec(R,u,[1,2,3]) %22 = y^6 + (-2.644862289350146893 E-38 + 0.E-38*I)*y^5 + (-6.0000000000000000000000000000000000001 + 0.E-37*I)*y^4 + (6.582768364604810045 E-37 - 3.526483052466862524 E-37*I)*y^3 + (9.0000000000000000000000000000000000029 + 2.4156408909398008289 E-36*I)*y^2 + (-2.2005254247393222149 E-35 + 4.713732346797372907 E-36*I)*y + (104.00000000000000000000000000000000001 + 0.E-35*I) (13:25) gp > substvec(R,u,[0,0,0]) %23 = y^6 (13:25) gp > substvec(R,u,[0,0,1]) %24 = y^6 + (0.E-38 + 0.E-38*I)*y^5 + (-2.0000000000000000000000000000000000000 + 0.E-38*I)*y^4 + (-4.0000000000000000000000000000000000000 + 0.E-38*I)*y^3 + (1.0000000000000000000000000000000000000 + 0.E-38*I)*y^2 + (4.0000000000000000000000000000000000000 + 0.E-37*I)*y + (4.0000000000000000000000000000000000000 + 0.E-38*I) (13:25) gp > round(%) %25 = y^6 - 2*y^4 - 4*y^3 + y^2 + 4*y + 4 (13:25) gp > factor(%) %26 = [y^3 - y - 2 2] (13:25) gp > P %27 = x^3 - x - 2 (13:26) gp > round(R) %28 = y^6 + (-2*u1^2 + (2*u2 + 2*u3)*u1 + (-2*u2^2 + 2*u3*u2 - 2*u3^2))*y^4 + (-4*u1^3 + (6*u2 + 6*u3)*u1^2 + (6*u2^2 - 24*u3*u2 + 6*u3^2)*u1 + (-4*u2^3 + 6*u3*u2^2 + 6*u3^2*u2 - 4*u3^3))*y^3 + (u1^4 + (-2*u2 - 2*u3)*u1^3 + (3*u2^2 + 3*u3^2)*u1^2 + (-2*u2^3 - 2*u3^3)*u1 + (u2^4 - 2*u3*u2^3 + 3*u3^2*u2^2 - 2*u3^3*u2 + u3^4))*y^2 + (4*u1^5 + (-10*u2 - 10*u3)*u1^4 + (4*u2^2 + 32*u3*u2 + 4*u3^2)*u1^3 + (4*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 4*u3^3)*u1^2 + (-10*u2^4 + 32*u3*u2^3 - 24*u3^2*u2^2 + 32*u3^3*u2 - 10*u3^4)*u1 + (4*u2^5 - 10*u3*u2^4 + 4*u3^2*u2^3 + 4*u3^3*u2^2 - 10*u3^4*u2 + 4*u3^5))*y + (4*u1^6 + (-12*u2 - 12*u3)*u1^5 + (23*u2^2 + 14*u3*u2 + 23*u3^2)*u1^4 + (-26*u2^3 - 14*u3*u2^2 - 14*u3^2*u2 - 26*u3^3)*u1^3 + (23*u2^4 - 14*u3*u2^3 + 42*u3^2*u2^2 - 14*u3^3*u2 + 23*u3^4)*u1^2 + (-12*u2^5 + 14*u3*u2^4 - 14*u3^2*u2^3 - 14*u3^3*u2^2 + 14*u3^4*u2 - 12*u3^5)*u1 + (4*u2^6 - 12*u3*u2^5 + 23*u3^2*u2^4 - 26*u3^3*u2^3 + 23*u3^4*u2^2 - 12*u3^5*u2 + 4*u3^6)) (13:26) gp > substvec(R,u,[1,2,3]) %29 = y^6 + (-2.644862289350146893 E-38 + 0.E-38*I)*y^5 + (-6.0000000000000000000000000000000000001 + 0.E-37*I)*y^4 + (6.582768364604810045 E-37 - 3.526483052466862524 E-37*I)*y^3 + (9.0000000000000000000000000000000000029 + 2.4156408909398008289 E-36*I)*y^2 + (-2.2005254247393222149 E-35 + 4.713732346797372907 E-36*I)*y + (104.00000000000000000000000000000000001 + 0.E-35*I) (13:26) gp > round(%) %30 = y^6 - 6*y^4 + 9*y^2 + 104 (13:26) gp > Q=% %31 = y^6 - 6*y^4 + 9*y^2 + 104 (13:26) gp > factor(Q) %32 = [y^6 - 6*y^4 + 9*y^2 + 104 1] (13:27) gp > polgalois(Q) %33 = [6, -1, 2, "D_6(6) = [3]2"] (13:27) gp > ?factormod factormod(f,{D},{flag=0}): factors the polynomial f over the finite field defined by the domain D; flag is optional, and can be 0: default or 1: only the degrees of the irreducible factors are given. (13:28) gp > ?nfsplitting nfsplitting(P,{d}): defining polynomial over Q for the splitting field of P, that is the smallest field over which P is totally split. P can also be given by a nf structure. If d is given, it must be a multiple of the splitting field degree. (13:28) gp > nfsplitting(Q) %34 = y^6 - 6*y^4 + 9*y^2 + 104 (13:28) gp > nfsplitting(P) %35 = x^6 - 6*x^4 + 9*x^2 + 104 (13:28) gp > R %36 = y^6 + ((-2.0571151139390031389 E-38 + 0.E-38*I)*u1 + ((-2.938735877055718770 E-39 + 0.E-38*I)*u2 + (0.E-38 + 0.E-38*I)*u3))*y^5 + ((-2.0000000000000000000000000000000000000 + 1.7632415262334312620 E-38*I)*u1^2 + ((2.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2 + (2.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u3)*u1 + ((-2.0000000000000000000000000000000000000 + 0.E-38*I)*u2^2 + (2.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (-2.0000000000000000000000000000000000000 + 0.E-38*I)*u3^2))*y^4 + ((-4.0000000000000000000000000000000000000 + 1.1754943508222875080 E-38*I)*u1^3 + ((6.0000000000000000000000000000000000000 + 0.E-37*I)*u2 + (6.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3)*u1^2 + ((6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u2^2 + (-24.000000000000000000000000000000000000 + 4.114230227878006278 E-38*I)*u3*u2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2)*u1 + ((-4.0000000000000000000000000000000000000 - 2.938735877055718770 E-39*I)*u2^3 + (6.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3*u2^2 + (6.0000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2*u2 + (-4.0000000000000000000000000000000000000 + 0.E-38*I)*u3^3))*y^3 + ((1.0000000000000000000000000000000000000 - 1.7632415262334312620 E-38*I)*u1^4 + ((-1.9999999999999999999999999999999999999 + 1.1754943508222875080 E-38*I)*u2 + (-2.0000000000000000000000000000000000001 + 4.701977403289150032 E-38*I)*u3)*u1^3 + ((2.9999999999999999999999999999999999999 + 0.E-37*I)*u2^2 + (4.701977403289150032 E-38 - 2.350988701644575016 E-38*I)*u3*u2 + (3.0000000000000000000000000000000000001 - 4.701977403289150032 E-38*I)*u3^2)*u1^2 + ((-1.9999999999999999999999999999999999999 - 3.526483052466862524 E-38*I)*u2^3 + (1.880790961315660013 E-37 + 9.403954806578300064 E-38*I)*u3*u2^2 + (-9.403954806578300064 E-38 - 5.289724578700293786 E-38*I)*u3^2*u2 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-37*I)*u3^3)*u1 + ((1.0000000000000000000000000000000000000 - 5.877471754111437540 E-39*I)*u2^4 + (-2.0000000000000000000000000000000000001 + 1.1754943508222875080 E-38*I)*u3*u2^3 + (3.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2*u2^2 + (-2.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3*u2 + (1.0000000000000000000000000000000000000 + 0.E-38*I)*u3^4))*y^2 + ((4.0000000000000000000000000000000000000 - 1.1754943508222875080 E-38*I)*u1^5 + ((-10.000000000000000000000000000000000000 + 9.403954806578300064 E-38*I)*u2 + (-10.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3)*u1^4 + ((3.9999999999999999999999999999999999999 + 9.403954806578300064 E-38*I)*u2^2 + (32.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (4.0000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u3^2)*u1^3 + ((4.0000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u2^3 + (-24.000000000000000000000000000000000000 - 9.403954806578300064 E-38*I)*u3*u2^2 + (-24.000000000000000000000000000000000000 - 1.880790961315660013 E-37*I)*u3^2*u2 + (4.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3)*u1^2 + ((-10.000000000000000000000000000000000000 - 4.701977403289150032 E-38*I)*u2^4 + (32.000000000000000000000000000000000000 + 7.052966104933725048 E-38*I)*u3*u2^3 + (-24.000000000000000000000000000000000000 + 1.6456920911512025112 E-37*I)*u3^2*u2^2 + (32.000000000000000000000000000000000000 + 9.403954806578300064 E-38*I)*u3^3*u2 + (-10.000000000000000000000000000000000000 - 1.1754943508222875080 E-37*I)*u3^4)*u1 + ((4.0000000000000000000000000000000000000 + 1.1754943508222875080 E-38*I)*u2^5 + (-10.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^4 + (4.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^3 + (4.0000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^3*u2^2 + (-10.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3^4*u2 + (4.0000000000000000000000000000000000000 + 0.E-37*I)*u3^5))*y + ((4.0000000000000000000000000000000000000 + 0.E-38*I)*u1^6 + ((-12.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u2 + (-12.000000000000000000000000000000000000 + 2.350988701644575016 E-38*I)*u3)*u1^5 + ((23.000000000000000000000000000000000000 + 4.701977403289150032 E-38*I)*u2^2 + (14.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (23.000000000000000000000000000000000000 - 2.350988701644575016 E-38*I)*u3^2)*u1^4 + ((-26.000000000000000000000000000000000000 + 0.E-37*I)*u2^3 + (-14.000000000000000000000000000000000000 - 1.4105932209867450096 E-37*I)*u3*u2^2 + (-14.000000000000000000000000000000000000 + 4.701977403289150032 E-38*I)*u3^2*u2 + (-26.000000000000000000000000000000000000 + 0.E-37*I)*u3^3)*u1^3 + ((23.000000000000000000000000000000000000 + 0.E-37*I)*u2^4 + (-14.000000000000000000000000000000000000 + 1.4105932209867450096 E-37*I)*u3*u2^3 + (42.00000000000000000[+++] (13:28) gp > round(R) %37 = y^6 + (-2*u1^2 + (2*u2 + 2*u3)*u1 + (-2*u2^2 + 2*u3*u2 - 2*u3^2))*y^4 + (-4*u1^3 + (6*u2 + 6*u3)*u1^2 + (6*u2^2 - 24*u3*u2 + 6*u3^2)*u1 + (-4*u2^3 + 6*u3*u2^2 + 6*u3^2*u2 - 4*u3^3))*y^3 + (u1^4 + (-2*u2 - 2*u3)*u1^3 + (3*u2^2 + 3*u3^2)*u1^2 + (-2*u2^3 - 2*u3^3)*u1 + (u2^4 - 2*u3*u2^3 + 3*u3^2*u2^2 - 2*u3^3*u2 + u3^4))*y^2 + (4*u1^5 + (-10*u2 - 10*u3)*u1^4 + (4*u2^2 + 32*u3*u2 + 4*u3^2)*u1^3 + (4*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 4*u3^3)*u1^2 + (-10*u2^4 + 32*u3*u2^3 - 24*u3^2*u2^2 + 32*u3^3*u2 - 10*u3^4)*u1 + (4*u2^5 - 10*u3*u2^4 + 4*u3^2*u2^3 + 4*u3^3*u2^2 - 10*u3^4*u2 + 4*u3^5))*y + (4*u1^6 + (-12*u2 - 12*u3)*u1^5 + (23*u2^2 + 14*u3*u2 + 23*u3^2)*u1^4 + (-26*u2^3 - 14*u3*u2^2 - 14*u3^2*u2 - 26*u3^3)*u1^3 + (23*u2^4 - 14*u3*u2^3 + 42*u3^2*u2^2 - 14*u3^3*u2 + 23*u3^4)*u1^2 + (-12*u2^5 + 14*u3*u2^4 - 14*u3^2*u2^3 - 14*u3^3*u2^2 + 14*u3^4*u2 - 12*u3^5)*u1 + (4*u2^6 - 12*u3*u2^5 + 23*u3^2*u2^4 - 26*u3^3*u2^3 + 23*u3^4*u2^2 - 12*u3^5*u2 + 4*u3^6)) (13:28) gp > R=% %38 = y^6 + (-2*u1^2 + (2*u2 + 2*u3)*u1 + (-2*u2^2 + 2*u3*u2 - 2*u3^2))*y^4 + (-4*u1^3 + (6*u2 + 6*u3)*u1^2 + (6*u2^2 - 24*u3*u2 + 6*u3^2)*u1 + (-4*u2^3 + 6*u3*u2^2 + 6*u3^2*u2 - 4*u3^3))*y^3 + (u1^4 + (-2*u2 - 2*u3)*u1^3 + (3*u2^2 + 3*u3^2)*u1^2 + (-2*u2^3 - 2*u3^3)*u1 + (u2^4 - 2*u3*u2^3 + 3*u3^2*u2^2 - 2*u3^3*u2 + u3^4))*y^2 + (4*u1^5 + (-10*u2 - 10*u3)*u1^4 + (4*u2^2 + 32*u3*u2 + 4*u3^2)*u1^3 + (4*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 4*u3^3)*u1^2 + (-10*u2^4 + 32*u3*u2^3 - 24*u3^2*u2^2 + 32*u3^3*u2 - 10*u3^4)*u1 + (4*u2^5 - 10*u3*u2^4 + 4*u3^2*u2^3 + 4*u3^3*u2^2 - 10*u3^4*u2 + 4*u3^5))*y + (4*u1^6 + (-12*u2 - 12*u3)*u1^5 + (23*u2^2 + 14*u3*u2 + 23*u3^2)*u1^4 + (-26*u2^3 - 14*u3*u2^2 - 14*u3^2*u2 - 26*u3^3)*u1^3 + (23*u2^4 - 14*u3*u2^3 + 42*u3^2*u2^2 - 14*u3^3*u2 + 23*u3^4)*u1^2 + (-12*u2^5 + 14*u3*u2^4 - 14*u3^2*u2^3 - 14*u3^3*u2^2 + 14*u3^4*u2 - 12*u3^5)*u1 + (4*u2^6 - 12*u3*u2^5 + 23*u3^2*u2^4 - 26*u3^3*u2^3 + 23*u3^4*u2^2 - 12*u3^5*u2 + 4*u3^6)) (13:29) gp > polcoeff(R,0,y) %39 = 4*u1^6 + (-12*u2 - 12*u3)*u1^5 + (23*u2^2 + 14*u3*u2 + 23*u3^2)*u1^4 + (-26*u2^3 - 14*u3*u2^2 - 14*u3^2*u2 - 26*u3^3)*u1^3 + (23*u2^4 - 14*u3*u2^3 + 42*u3^2*u2^2 - 14*u3^3*u2 + 23*u3^4)*u1^2 + (-12*u2^5 + 14*u3*u2^4 - 14*u3^2*u2^3 - 14*u3^3*u2^2 + 14*u3^4*u2 - 12*u3^5)*u1 + (4*u2^6 - 12*u3*u2^5 + 23*u3^2*u2^4 - 26*u3^3*u2^3 + 23*u3^4*u2^2 - 12*u3^5*u2 + 4*u3^6) (13:29) gp > polcoeff(R,6,y) %40 = 1 (13:29) gp > polcoeff(R,5,y) %41 = 0 (13:29) gp > polcoeff(R,4,y) %42 = -2*u1^2 + (2*u2 + 2*u3)*u1 + (-2*u2^2 + 2*u3*u2 - 2*u3^2) (13:29) gp > polciclo(7) *** at top-level: polciclo(7) *** ^----------- *** not a function in function call *** Break loop: type 'break' to go back to GP prompt break> break (13:37) gp > polcyclo(7) %43 = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 (13:37) gp > %/x^3 %44 = (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/x^3 (13:37) gp > algdep(cos(2*Pi/7)) *** too few arguments: algdep(cos(2*Pi/7)) *** ^- (13:37) gp > algdep(cos(2*Pi/7),3) %45 = 8*x^3 + 4*x^2 - 4*x - 1 (13:37) gp > subst(%,x,x/2) %46 = x^3 + x^2 - 2*x - 1 (13:37) gp > P=% %47 = x^3 + x^2 - 2*x - 1 (13:37) gp > factor(P) %48 = [x^3 + x^2 - 2*x - 1 1] (13:38) gp > polgalois(P) %49 = [3, 1, 1, "A3"] (13:38) gp > X=polroots(P) %50 = [-1.8019377358048382524722046390148901023 + 0.E-38*I, -0.44504186791262880857780512899358951893 + 0.E-38*I, 1.2469796037174670610500097680084796213 + 0.E-38*I]~ (13:38) gp > R=1;forperm(3,s,R=R*(y-sum(n=1,poldegree(P),X[s[n]]*u[n]))) (13:38) gp > R %52 = y^6 + ((2.0000000000000000000000000000000000000 + 0.E-38*I)*u1 + ((2.0000000000000000000000000000000000000 + 0.E-38*I)*u2 + (2.0000000000000000000000000000000000000 + 0.E-38*I)*u3))*y^5 + ((-3.0000000000000000000000000000000000000 + 0.E-37*I)*u1^2 + ((7.9999999999999999999999999999999999999 + 0.E-37*I)*u2 + (8.0000000000000000000000000000000000000 + 0.E-37*I)*u3)*u1 + ((-3.0000000000000000000000000000000000000 + 0.E-38*I)*u2^2 + (8.0000000000000000000000000000000000000 + 0.E-38*I)*u3*u2 + (-3.0000000000000000000000000000000000000 + 0.E-38*I)*u3^2))*y^4 + ((-6.0000000000000000000000000000000000000 + 0.E-37*I)*u1^3 + ((3.0000000000000000000000000000000000001 + 0.E-37*I)*u2 + (3.0000000000000000000000000000000000001 + 0.E-37*I)*u3)*u1^2 + ((3.0000000000000000000000000000000000000 + 0.E-37*I)*u2^2 + (20.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (3.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2)*u1 + ((-6.0000000000000000000000000000000000000 + 0.E-38*I)*u2^3 + (2.9999999999999999999999999999999999999 + 0.E-37*I)*u3*u2^2 + (3.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2 + (-6.0000000000000000000000000000000000000 + 0.E-38*I)*u3^3))*y^3 + ((2.0000000000000000000000000000000000001 + 0.E-37*I)*u1^4 + ((-13.000000000000000000000000000000000000 + 0.E-37*I)*u2 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3)*u1^3 + ((19.000000000000000000000000000000000000 + 0.E-37*I)*u2^2 + (9.9999999999999999999999999999999999999 + 0.E-37*I)*u3*u2 + (19.000000000000000000000000000000000000 + 0.E-37*I)*u3^2)*u1^2 + ((-13.000000000000000000000000000000000000 + 0.E-37*I)*u2^3 + (10.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^2 + (10.000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3^3)*u1 + ((2.0000000000000000000000000000000000000 + 0.E-37*I)*u2^4 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^3 + (19.000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^2 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3^3*u2 + (2.0000000000000000000000000000000000000 + 0.E-38*I)*u3^4))*y^2 + ((4.0000000000000000000000000000000000000 + 0.E-37*I)*u1^5 + ((-8.0000000000000000000000000000000000000 + 0.E-37*I)*u2 + (-7.9999999999999999999999999999999999999 + 0.E-37*I)*u3)*u1^4 + ((5.0000000000000000000000000000000000000 + 0.E-37*I)*u2^2 + (-4.0000000000000000000000000000000000001 + 0.E-37*I)*u3*u2 + (4.9999999999999999999999999999999999997 + 0.E-37*I)*u3^2)*u1^3 + ((5.0000000000000000000000000000000000001 + 0.E-37*I)*u2^3 + (7.9999999999999999999999999999999999998 + 0.E-37*I)*u3*u2^2 + (8.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2 + (5.0000000000000000000000000000000000001 + 0.E-37*I)*u3^3)*u1^2 + ((-7.9999999999999999999999999999999999999 + 0.E-37*I)*u2^4 + (-4.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^3 + (8.0000000000000000000000000000000000001 + 0.E-37*I)*u3^2*u2^2 + (-4.0000000000000000000000000000000000000 + 0.E-37*I)*u3^3*u2 + (-8.0000000000000000000000000000000000000 + 0.E-37*I)*u3^4)*u1 + ((4.0000000000000000000000000000000000000 + 0.E-38*I)*u2^5 + (-8.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^4 + (5.0000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^3 + (4.9999999999999999999999999999999999999 + 0.E-37*I)*u3^3*u2^2 + (-7.9999999999999999999999999999999999999 + 0.E-37*I)*u3^4*u2 + (4.0000000000000000000000000000000000000 + 0.E-38*I)*u3^5))*y + ((1.0000000000000000000000000000000000000 + 0.E-38*I)*u1^6 + ((-0.99999999999999999999999999999999999999 + 0.E-37*I)*u2 + (-1.0000000000000000000000000000000000000 + 0.E-37*I)*u3)*u1^5 + ((-13.000000000000000000000000000000000000 + 0.E-37*I)*u2^2 + (23.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3^2)*u1^4 + ((27.000000000000000000000000000000000000 + 0.E-37*I)*u2^3 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^2 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2 + (27.000000000000000000000000000000000000 + 0.E-37*I)*u3^3)*u1^3 + ((-13.000000000000000000000000000000000000 + 0.E-37*I)*u2^4 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^3 + (75.999999999999999999999999999999999999 + 0.E-37*I)*u3^2*u2^2 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3^3*u2 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3^4)*u1^2 + ((-0.99999999999999999999999999999999999999 + 0.E-37*I)*u2^5 + (23.000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^4 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^3 + (-24.000000000000000000000000000000000000 + 0.E-37*I)*u3^3*u2^2 + (23.000000000000000000000000000000000000 + 0.E-37*I)*u3^4*u2 + (-0.99999999999999999999999999999999999999 + 0.E-38*I)*u3^5)*u1 + ((1.0000000000000000000000000000000000000 + 0.E-38*I)*u2^6 + (-1.0000000000000000000000000000000000000 + 0.E-37*I)*u3*u2^5 + (-13.000000000000000000000000000000000000 + 0.E-37*I)*u3^2*u2^4 + (27.000000000000000000000000000000000000 + 0.E-37*I)*u3^3*u2^3 + (-13.00[+++] (13:38) gp > round(R) %53 = y^6 + (2*u1 + (2*u2 + 2*u3))*y^5 + (-3*u1^2 + (8*u2 + 8*u3)*u1 + (-3*u2^2 + 8*u3*u2 - 3*u3^2))*y^4 + (-6*u1^3 + (3*u2 + 3*u3)*u1^2 + (3*u2^2 + 20*u3*u2 + 3*u3^2)*u1 + (-6*u2^3 + 3*u3*u2^2 + 3*u3^2*u2 - 6*u3^3))*y^3 + (2*u1^4 + (-13*u2 - 13*u3)*u1^3 + (19*u2^2 + 10*u3*u2 + 19*u3^2)*u1^2 + (-13*u2^3 + 10*u3*u2^2 + 10*u3^2*u2 - 13*u3^3)*u1 + (2*u2^4 - 13*u3*u2^3 + 19*u3^2*u2^2 - 13*u3^3*u2 + 2*u3^4))*y^2 + (4*u1^5 + (-8*u2 - 8*u3)*u1^4 + (5*u2^2 - 4*u3*u2 + 5*u3^2)*u1^3 + (5*u2^3 + 8*u3*u2^2 + 8*u3^2*u2 + 5*u3^3)*u1^2 + (-8*u2^4 - 4*u3*u2^3 + 8*u3^2*u2^2 - 4*u3^3*u2 - 8*u3^4)*u1 + (4*u2^5 - 8*u3*u2^4 + 5*u3^2*u2^3 + 5*u3^3*u2^2 - 8*u3^4*u2 + 4*u3^5))*y + (u1^6 + (-u2 - u3)*u1^5 + (-13*u2^2 + 23*u3*u2 - 13*u3^2)*u1^4 + (27*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 27*u3^3)*u1^3 + (-13*u2^4 - 24*u3*u2^3 + 76*u3^2*u2^2 - 24*u3^3*u2 - 13*u3^4)*u1^2 + (-u2^5 + 23*u3*u2^4 - 24*u3^2*u2^3 - 24*u3^3*u2^2 + 23*u3^4*u2 - u3^5)*u1 + (u2^6 - u3*u2^5 - 13*u3^2*u2^4 + 27*u3^3*u2^3 - 13*u3^4*u2^2 - u3^5*u2 + u3^6)) (13:38) gp > R=round(R) %54 = y^6 + (2*u1 + (2*u2 + 2*u3))*y^5 + (-3*u1^2 + (8*u2 + 8*u3)*u1 + (-3*u2^2 + 8*u3*u2 - 3*u3^2))*y^4 + (-6*u1^3 + (3*u2 + 3*u3)*u1^2 + (3*u2^2 + 20*u3*u2 + 3*u3^2)*u1 + (-6*u2^3 + 3*u3*u2^2 + 3*u3^2*u2 - 6*u3^3))*y^3 + (2*u1^4 + (-13*u2 - 13*u3)*u1^3 + (19*u2^2 + 10*u3*u2 + 19*u3^2)*u1^2 + (-13*u2^3 + 10*u3*u2^2 + 10*u3^2*u2 - 13*u3^3)*u1 + (2*u2^4 - 13*u3*u2^3 + 19*u3^2*u2^2 - 13*u3^3*u2 + 2*u3^4))*y^2 + (4*u1^5 + (-8*u2 - 8*u3)*u1^4 + (5*u2^2 - 4*u3*u2 + 5*u3^2)*u1^3 + (5*u2^3 + 8*u3*u2^2 + 8*u3^2*u2 + 5*u3^3)*u1^2 + (-8*u2^4 - 4*u3*u2^3 + 8*u3^2*u2^2 - 4*u3^3*u2 - 8*u3^4)*u1 + (4*u2^5 - 8*u3*u2^4 + 5*u3^2*u2^3 + 5*u3^3*u2^2 - 8*u3^4*u2 + 4*u3^5))*y + (u1^6 + (-u2 - u3)*u1^5 + (-13*u2^2 + 23*u3*u2 - 13*u3^2)*u1^4 + (27*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 27*u3^3)*u1^3 + (-13*u2^4 - 24*u3*u2^3 + 76*u3^2*u2^2 - 24*u3^3*u2 - 13*u3^4)*u1^2 + (-u2^5 + 23*u3*u2^4 - 24*u3^2*u2^3 - 24*u3^3*u2^2 + 23*u3^4*u2 - u3^5)*u1 + (u2^6 - u3*u2^5 - 13*u3^2*u2^4 + 27*u3^3*u2^3 - 13*u3^4*u2^2 - u3^5*u2 + u3^6)) (13:38) gp > substvec(R,u,[1,2,3]) %55 = y^6 + 12*y^5 + 46*y^4 + 48*y^3 - 47*y^2 - 60*y - 13 (13:38) gp > factor(%) %56 = [y^3 + 6*y^2 + 5*y - 13 1] [ y^3 + 6*y^2 + 5*y + 1 1] (13:38) gp > substvec(R,u,[1,11,111]) %57 = y^6 + 246*y^5 - 26585*y^4 - 7704780*y^3 + 118417615*y^2 + 53685042006*y + 1508089946441 (13:39) gp > factor(%0 *** syntax error, unexpected $end, expecting )-> or ',' or ')': factor(%0 *** ^- (13:39) gp > factor(%) %58 = [y^3 + 123*y^2 - 20857*y - 1671979 1] [ y^3 + 123*y^2 - 20857*y - 901979 1] (13:39) gp > R %59 = y^6 + (2*u1 + (2*u2 + 2*u3))*y^5 + (-3*u1^2 + (8*u2 + 8*u3)*u1 + (-3*u2^2 + 8*u3*u2 - 3*u3^2))*y^4 + (-6*u1^3 + (3*u2 + 3*u3)*u1^2 + (3*u2^2 + 20*u3*u2 + 3*u3^2)*u1 + (-6*u2^3 + 3*u3*u2^2 + 3*u3^2*u2 - 6*u3^3))*y^3 + (2*u1^4 + (-13*u2 - 13*u3)*u1^3 + (19*u2^2 + 10*u3*u2 + 19*u3^2)*u1^2 + (-13*u2^3 + 10*u3*u2^2 + 10*u3^2*u2 - 13*u3^3)*u1 + (2*u2^4 - 13*u3*u2^3 + 19*u3^2*u2^2 - 13*u3^3*u2 + 2*u3^4))*y^2 + (4*u1^5 + (-8*u2 - 8*u3)*u1^4 + (5*u2^2 - 4*u3*u2 + 5*u3^2)*u1^3 + (5*u2^3 + 8*u3*u2^2 + 8*u3^2*u2 + 5*u3^3)*u1^2 + (-8*u2^4 - 4*u3*u2^3 + 8*u3^2*u2^2 - 4*u3^3*u2 - 8*u3^4)*u1 + (4*u2^5 - 8*u3*u2^4 + 5*u3^2*u2^3 + 5*u3^3*u2^2 - 8*u3^4*u2 + 4*u3^5))*y + (u1^6 + (-u2 - u3)*u1^5 + (-13*u2^2 + 23*u3*u2 - 13*u3^2)*u1^4 + (27*u2^3 - 24*u3*u2^2 - 24*u3^2*u2 + 27*u3^3)*u1^3 + (-13*u2^4 - 24*u3*u2^3 + 76*u3^2*u2^2 - 24*u3^3*u2 - 13*u3^4)*u1^2 + (-u2^5 + 23*u3*u2^4 - 24*u3^2*u2^3 - 24*u3^3*u2^2 + 23*u3^4*u2 - u3^5)*u1 + (u2^6 - u3*u2^5 - 13*u3^2*u2^4 + 27*u3^3*u2^3 - 13*u3^4*u2^2 - u3^5*u2 + u3^6)) (13:39) gp > factor(R) *** at top-level: factor(R) *** ^--------- *** factor: user interrupt after 4,314 ms *** Break loop: to continue; 'break' to go back to GP prompt break> break (13:39) gp > substvec(R,u,[0,0,v]) %60 = [y^6 + 2*y^5 - 3*y^4 - 6*y^3 + 2*y^2 + 4*y + 1, y^6 + 4*y^5 - 12*y^4 - 48*y^3 + 32*y^2 + 128*y + 64, y^6 + 6*y^5 - 27*y^4 - 162*y^3 + 162*y^2 + 972*y + 729] (13:39) gp > v %61 = [1, 2, 3] (13:39) gp > substvec(R,u,v) %62 = y^6 + 12*y^5 + 46*y^4 + 48*y^3 - 47*y^2 - 60*y - 13 (13:40) gp > substvec(R,u,[0,0,z]) %63 = y^6 + 2*z*y^5 - 3*z^2*y^4 - 6*z^3*y^3 + 2*z^4*y^2 + 4*z^5*y + z^6 (13:40) gp > factor(%) %64 = [y^3 + z*y^2 - 2*z^2*y - z^3 2] (13:40) gp > substvec(R,u,[0,1,z]) %65 = y^6 + (2*z + 2)*y^5 + (-3*z^2 + 8*z - 3)*y^4 + (-6*z^3 + 3*z^2 + 3*z - 6)*y^3 + (2*z^4 - 13*z^3 + 19*z^2 - 13*z + 2)*y^2 + (4*z^5 - 8*z^4 + 5*z^3 + 5*z^2 - 8*z + 4)*y + (z^6 - z^5 - 13*z^4 + 27*z^3 - 13*z^2 - z + 1) (13:40) gp > factor(%) %66 = [y^3 + (z + 1)*y^2 + (-2*z^2 + 3*z - 2)*y + (-z^3 + 4*z^2 - 3*z - 1) 1] [y^3 + (z + 1)*y^2 + (-2*z^2 + 3*z - 2)*y + (-z^3 - 3*z^2 + 4*z - 1) 1] (13:40) gp > substvec(R,u,[1,z,z^2]) %67 = y^6 + (2*z^2 + 2*z + 2)*y^5 + (-3*z^4 + 8*z^3 + 5*z^2 + 8*z - 3)*y^4 + (-6*z^6 + 3*z^5 + 6*z^4 + 14*z^3 + 6*z^2 + 3*z - 6)*y^3 + (2*z^8 - 13*z^7 + 6*z^6 - 3*z^5 + 31*z^4 - 3*z^3 + 6*z^2 - 13*z + 2)*y^2 + (4*z^10 - 8*z^9 - 3*z^8 + z^7 + 5*z^6 + 8*z^5 + 5*z^4 + z^3 - 3*z^2 - 8*z + 4)*y + (z^12 - z^11 - 14*z^10 + 50*z^9 - 50*z^8 - 49*z^7 + 127*z^6 - 49*z^5 - 50*z^4 + 50*z^3 - 14*z^2 - z + 1) (13:40) gp > factor(%) %68 = [y^3 + (z^2 + z + 1)*y^2 + (-2*z^4 + 3*z^3 + z^2 + 3*z - 2)*y + (-z^6 + 4*z^5 - 6*z^4 + 8*z^2 - 3*z - 1) 1] [y^3 + (z^2 + z + 1)*y^2 + (-2*z^4 + 3*z^3 + z^2 + 3*z - 2)*y + (-z^6 - 3*z^5 + 8*z^4 - 6*z^2 + 4*z - 1) 1] (13:40) gp > substvec(R,u,[1,z,w]) %69 = y^6 + (2*z + (2*w + 2))*y^5 + (-3*z^2 + (8*w + 8)*z + (-3*w^2 + 8*w - 3))*y^4 + (-6*z^3 + (3*w + 3)*z^2 + (3*w^2 + 20*w + 3)*z + (-6*w^3 + 3*w^2 + 3*w - 6))*y^3 + (2*z^4 + (-13*w - 13)*z^3 + (19*w^2 + 10*w + 19)*z^2 + (-13*w^3 + 10*w^2 + 10*w - 13)*z + (2*w^4 - 13*w^3 + 19*w^2 - 13*w + 2))*y^2 + (4*z^5 + (-8*w - 8)*z^4 + (5*w^2 - 4*w + 5)*z^3 + (5*w^3 + 8*w^2 + 8*w + 5)*z^2 + (-8*w^4 - 4*w^3 + 8*w^2 - 4*w - 8)*z + (4*w^5 - 8*w^4 + 5*w^3 + 5*w^2 - 8*w + 4))*y + (z^6 + (-w - 1)*z^5 + (-13*w^2 + 23*w - 13)*z^4 + (27*w^3 - 24*w^2 - 24*w + 27)*z^3 + (-13*w^4 - 24*w^3 + 76*w^2 - 24*w - 13)*z^2 + (-w^5 + 23*w^4 - 24*w^3 - 24*w^2 + 23*w - 1)*z + (w^6 - w^5 - 13*w^4 + 27*w^3 - 13*w^2 - w + 1)) (13:40) gp > factor(%) %70 = [y^3 + (z + (w + 1))*y^2 + (-2*z^2 + (3*w + 3)*z + (-2*w^2 + 3*w - 2))*y + (-z^3 + (4*w - 3)*z^2 + (-3*w^2 + w + 4)*z + (-w^3 + 4*w^2 - 3*w - 1)) 1] [y^3 + (z + (w + 1))*y^2 + (-2*z^2 + (3*w + 3)*z + (-2*w^2 + 3*w - 2))*y + (-z^3 + (-3*w + 4)*z^2 + (4*w^2 + w - 3)*z + (-w^3 - 3*w^2 + 4*w - 1)) 1] (13:40) gp > factor(R) *** factor: Warning: increasing stack size to 80000000. *** factor: Warning: increasing stack size to 160000000. *** factor: Warning: increasing stack size to 320000000. *** at top-level: factor(R) *** ^--------- *** factor: user interrupt after 28min, 57,912 ms *** Break loop: to continue; 'break' to go back to GP prompt break> break (14:10) gp > P=x^5-x-1 %71 = x^5 - x - 1 (14:10) gp > facrtormod(P,2) *** at top-level: facrtormod(P,2) *** ^--------------- *** not a function in function call *** Break loop: type 'break' to go back to GP prompt break> break (14:10) gp > factormod(P,2) %72 = [ Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2) 1] [Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2) 1] (14:10) gp > factor(P) %73 = [x^5 - x - 1 1] (14:10) gp > "Então Gal(P) contém Z/6 = Z/2 x Z/3" %74 = "Então Gal(P) contém Z/6 = Z/2 x Z/3" (14:11) gp > factormod(P,3) %75 = [Mod(1, 3)*x^5 + Mod(2, 3)*x + Mod(2, 3) 1] (14:11) gp > "Então Gal(P) contém Z/5 - um 5-ciclo" %76 = "Então Gal(P) contém Z/5 - um 5-ciclo" (14:12) gp > "Agora usando teoria de permutações podemos mostrar que Gal(P) deve ser todo o S_5" %77 = "Agora usando teoria de permutações podemos mostrar que Gal(P) deve ser todo o S_5" (14:12) gp > factormod(P,5) %78 = [Mod(1, 5)*x^5 + Mod(4, 5)*x + Mod(4, 5) 1] (14:14) gp > factormod(P,7) %79 = [ Mod(1, 7)*x^2 + Mod(6, 7)*x + Mod(3, 7) 1] [Mod(1, 7)*x^3 + Mod(1, 7)*x^2 + Mod(5, 7)*x + Mod(2, 7) 1] (14:14) gp > factormod(P,11) %80 = [Mod(1, 11)*x^5 + Mod(10, 11)*x + Mod(10, 11) 1] (14:14) gp > factormod(P,13) %81 = [Mod(1, 13)*x^5 + Mod(12, 13)*x + Mod(12, 13) 1] (14:14) gp > factormod(P,17) %82 = [ Mod(1, 17)*x + Mod(9, 17) 1] [ Mod(1, 17)*x + Mod(11, 17) 1] [Mod(1, 17)*x^3 + Mod(14, 17)*x^2 + Mod(12, 17)*x + Mod(6, 17) 1] (14:14) gp > factormod(P,23) %83 = [ Mod(1, 23)*x + Mod(9, 23) 1] [Mod(1, 23)*x^4 + Mod(14, 23)*x^3 + Mod(12, 23)*x^2 + Mod(7, 23)*x + Mod(5, 23) 1] (14:15) gp > factormod(P,2) %84 = [ Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2) 1] [Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2) 1] (14:15) gp > factormod(P,3) %85 = [Mod(1, 3)*x^5 + Mod(2, 3)*x + Mod(2, 3) 1] (14:15) gp > factormod(P,23) %86 = [ Mod(1, 23)*x + Mod(9, 23) 1] [Mod(1, 23)*x^4 + Mod(14, 23)*x^3 + Mod(12, 23)*x^2 + Mod(7, 23)*x + Mod(5, 23) 1] (14:15) gp > lcm([6,5,4]) %87 = 60 (14:15) gp > poldisc(P) %88 = 2869 (14:17) gp > factor(poldisc(P)) %89 = [ 19 1] [151 1] (14:17) gp > polgalois(x^5-x-3) %90 = [120, -1, 1, "S5"] (14:17) gp > polgalois(x^5-x-2) %91 = [120, -1, 1, "S5"] (14:17) gp > "Então se olhamos por fatorações de polinómio x^5-x-1 módulo primos 2, 3 e 23 + mostramos que Disc(P) não é quadrado => provamos que Gal(P) é Sym(5)" %92 = "Então se olhamos por fatorações de polinómio x^5-x-1 módulo primos 2, 3 e 23 + mostramos que Disc(P) não é quadrado => provamos que Gal(P) é Sym(5)" Logging to /Users/s/tmp/pari-12.23 GPRC Done. GP/PARI CALCULATOR Version 2.13.0 (released) i386 running darwin (x86-64/GMP-6.2.0 kernel) 64-bit version compiled: Oct 31 2020, Apple clang version 12.0.0 (clang-1200.0.32.21) threading engine: single (readline v8.0 enabled, extended help enabled) Copyright (C) 2000-2020 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisizemax = 900001792, primelimit = 1000000 (15:38) gp > algdep(cos(2*Pi/11),5) %1 = 32*x^5 + 16*x^4 - 32*x^3 - 12*x^2 + 6*x + 1 (15:38) gp > subst(%,x,x/2) %2 = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1 (15:38) gp > P=% %3 = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1 (15:38) gp > polgalois(P) %4 = [5, 1, 1, "C(5) = 5"] (15:38) gp > factormod(P,2) %5 = [Mod(1, 2)*x^5 + Mod(1, 2)*x^4 + Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2) 1] (15:39) gp > factormod(P,3) %6 = [Mod(1, 3)*x^5 + Mod(1, 3)*x^4 + Mod(2, 3)*x^3 + Mod(1, 3) 1] (15:39) gp > factormod(P,5) %7 = [Mod(1, 5)*x^5 + Mod(1, 5)*x^4 + Mod(1, 5)*x^3 + Mod(2, 5)*x^2 + Mod(3, 5)*x + Mod(1, 5) 1] (15:39) gp > factormod(P,7) %8 = [Mod(1, 7)*x^5 + Mod(1, 7)*x^4 + Mod(3, 7)*x^3 + Mod(4, 7)*x^2 + Mod(3, 7)*x + Mod(1, 7) 1] (15:39) gp > factormod(P,11) %9 = [Mod(1, 11)*x + Mod(9, 11) 5] (15:39) gp > factormod(P,13) %10 = [Mod(1, 13)*x^5 + Mod(1, 13)*x^4 + Mod(9, 13)*x^3 + Mod(10, 13)*x^2 + Mod(3, 13)*x + Mod(1, 13) 1] (15:39) gp > factormod(P,17) %11 = [Mod(1, 17)*x^5 + Mod(1, 17)*x^4 + Mod(13, 17)*x^3 + Mod(14, 17)*x^2 + Mod(3, 17)*x + Mod(1, 17) 1] (15:39) gp > factormod(P,19) %12 = [Mod(1, 19)*x^5 + Mod(1, 19)*x^4 + Mod(15, 19)*x^3 + Mod(16, 19)*x^2 + Mod(3, 19)*x + Mod(1, 19) 1] (15:39) gp > factormod(P,23) %13 = [ Mod(1, 23)*x + Mod(9, 23) 1] [Mod(1, 23)*x + Mod(12, 23) 1] [Mod(1, 23)*x + Mod(13, 23) 1] [Mod(1, 23)*x + Mod(17, 23) 1] [Mod(1, 23)*x + Mod(19, 23) 1] (15:39) gp > poldisc(P) %14 = 14641 (15:39) gp > factor(%) %15 = [11 4] (15:39) gp > factormod(P,29) %16 = [Mod(1, 29)*x^5 + Mod(1, 29)*x^4 + Mod(25, 29)*x^3 + Mod(26, 29)*x^2 + Mod(3, 29)*x + Mod(1, 29) 1] (15:40) gp > factormod(P,31) %17 = [Mod(1, 31)*x^5 + Mod(1, 31)*x^4 + Mod(27, 31)*x^3 + Mod(28, 31)*x^2 + Mod(3, 31)*x + Mod(1, 31) 1] (15:40) gp > factormod(P,41) %18 = [Mod(1, 41)*x^5 + Mod(1, 41)*x^4 + Mod(37, 41)*x^3 + Mod(38, 41)*x^2 + Mod(3, 41)*x + Mod(1, 41) 1] (15:40) gp > factormod(P,43) %19 = [ Mod(1, 43)*x + Mod(7, 43) 1] [Mod(1, 43)*x + Mod(21, 43) 1] [Mod(1, 43)*x + Mod(29, 43) 1] [Mod(1, 43)*x + Mod(34, 43) 1] [Mod(1, 43)*x + Mod(39, 43) 1] (15:40) gp > P=x^5-x^4+2*x^3-4*x^2+x-1 %20 = x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1 (15:51) gp > factor(P) %21 = [x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1 1] (15:51) gp > polgalois(P) %22 = [20, -1, 1, "F(5) = 5:4"] (15:51) gp > polroots(P) %23 = [1.4391720150751641173298092304417563098 + 0.E-38*I, 0.078181872114742299904698394449057706660 - 0.52783322138609194439109621984797033387*I, 0.078181872114742299904698394449057706660 + 0.52783322138609194439109621984797033387*I, -0.29776787965232435856960300966993586156 - 1.5335508632480834995476816634577257310*I, -0.29776787965232435856960300966993586156 + 1.5335508632480834995476816634577257310*I]~ (15:51) gp > Q=x^5-9*x^3-4*x^2+17*x+12 %24 = x^5 - 9*x^3 - 4*x^2 + 17*x + 12 (15:51) gp > polgalois(Q) %25 = [20, -1, 1, "F(5) = 5:4"] (15:51) gp > polroots(Q) %26 = [-2.1766392753506720687180759370903410742 + 0.E-38*I, -1.4552730944755529730079541428404195057 + 0.E-38*I, -0.81507913996118546183745835334303919268 + 0.E-38*I, 1.6793446099735060222379333362773123397 + 0.E-38*I, 2.7676468998139044813255550969964874328 + 0.E-38*I]~ (15:52) gp > real(%) %27 = [-2.1766392753506720687180759370903410742, -1.4552730944755529730079541428404195057, -0.81507913996118546183745835334303919268, 1.6793446099735060222379333362773123397, 2.7676468998139044813255550969964874328]~ (15:52) gp > X=% %28 = [-2.1766392753506720687180759370903410742, -1.4552730944755529730079541428404195057, -0.81507913996118546183745835334303919268, 1.6793446099735060222379333362773123397, 2.7676468998139044813255550969964874328]~ (15:52) gp > u=[a,b,c,d,e] %29 = [a, b, c, d, e] (15:52) gp > R=1 %30 = 1 (15:52) gp > forperm(5,s,R*=1) (15:52) gp > R %32 = 1 (15:52) gp > forperm(5,s,R*=(y+sum(n=1,5,X[s[n]]*u[n]))) *** _*=_: Warning: increasing stack size to 80000000. *** _*=_: Warning: increasing stack size to 160000000. *** _*=_: Warning: increasing stack size to 320000000. *** at top-level: forperm(5,s,R*=(y+sum(n=1,5,X[s[n]]*u[n]))) *** ^------------------------------ *** _*=_: user interrupt after 25,752 ms *** Break loop: to continue; 'break' to go back to GP prompt break> break (15:53) gp > u=[1,2,3,4,5] %33 = [1, 2, 3, 4, 5] (15:53) gp > forperm(5,s,R*=(y+sum(n=1,5,X[s[n]]*u[n]))) (15:53) gp > R %35 = 0 (15:54) gp > R=1 %36 = 1 (15:54) gp > forperm(5,s,R*=(y+sum(n=1,5,X[s[n]]*u[n]))) (15:54) gp > R %38 = y^120 - 2.350988701644575016 E-37*y^119 - 2699.9999999999999999999999999999999999*y^118 + 5.238529448733281520 E-32*y^117 + 3515489.9999999999999999999999999999870*y^116 + 1.5643111750532664101 E-27*y^115 - 2941059000.0000000000000000000000001310*y^114 + 8.809796641116523264 E-24*y^113 + 1777297395054.9999999999999999999994797*y^112 + 2.849812986125347858 E-20*y^111 - 826923180241800.00000000000000000145895*y^110 + 6.800748124165972332 E-17*y^109 + 308333101945192509.99999999999999719622*y^108 + 1.0070996770902840289 E-13*y^107 - 94694043847528345100.000000000003149838*y^106 + 8.656301717091530179 E-11*y^105 + 24433237971035864924274.999999997882057*y^104 + 4.660477692652875703 E-8*y^103 - 5376233160316134541531400.0000009202484*y^102 + 1.5882219486229587346 E-5*y^101 + 1020567781879348259569748623.9997777123*y^100 + 0.0019386339990887790918*y^99 - 168678245702960424035497928999.98818035*y^98 - 1.0034765414893627167*y^97 + 24453685182038118959187459147347.447693*y^96 - 541.2671923637390137*y^95 - 3128400158918188453168932998285337.8199*y^94 - 117024.16583251953125*y^93 + 354940729606632175277410832806792530.05*y^92 - 13450679.820312500000*y^91 - 3.5862297516226338578321083235136069159 E37*y^90 - 772859863.5000000000*y^89 + 3.2378800620413304418416256667784405017 E39*y^88 - 3864126080.000000000*y^87 - 2.6197818543849202252322918265833876720 E41*y^86 + 2542440812544.000000*y^85 + 1.9040635071508774613919329587704629657 E43*y^84 + 162819757572096.00000*y^83 - 1.2455438225439111347606343673801939134 E45*y^82 + 2252061538254848.000*y^81 + 7.3449702876006956019459920354868083728 E46*y^80 - 227800548147986432.00*y^79 - 3.9095618379266916049630110762547996218 E48*y^78 - 1.3332472321017905152 E19*y^77 + 1.8801943272860258946991790689221597449 E50*y^76 - 2.1468610385456555622 E20*y^75 - 8.1758160715823204864091109291243878685 E51*y^74 + 5.366666926684987458 E21*y^73 + 3.2160795372966689825055635796624051923 E53*y^72 + 2.9699080742027041072 E23*y^71 - 1.1447392587462808219222815584010652525 E55*y^70 + 4.071629771438242140 E24*y^69 + 3.6871907306675683333554744835838622687 E56*y^68 - 3.966606145181375941 E25*y^67 - 1.0745599367839809852079685586236589924 E58*y^66 - 1.4973677430273782198 E27*y^65 + 2.8324493826515329893166726489746612988 E59*y^64 + 1.6538671896306071467 E27*y^63 - 6.7492408516090205629893020107443036743 E60*y^62 + 3.573700326312344489 E29*y^61 + 1.4527384772638614736350070128768961651 E62*y^60 + 2.4107594759084891274 E30*y^59 - 2.8219544014787192200882721406049352109 E63*y^58 - 1.3044851192409036149 E31*y^57 + 4.9413030563043764839681129994624267467 E64*y^56 - 1.6800053324640913576 E32*y^55 - 7.7886983587398879502079321871591096783 E65*y^54 - 1.1515516315838892978 E33*y^53 + 1.1033698687890523199903713340701818780 E67*y^52 + 8.980512220991862666 E32*y^51 - 1.4021725300996717272541490822329323171 E68*y^50 + 1.5122247687832628410 E35*y^49 + 1.5950633476388864378600546833040758053 E69*y^48 + 2.2155997190788993478 E35*y^47 - 1.6202862866651499804697854215400464690 E70*y^46 - 5.989273861500716366 E36*y^45 + 1.4656799374488157967366144177323700980 E71*y^44 + 3.724823958891421970 E36*y^43 - 1.1769399599911437832088914888408914298 E72*[+++] (15:54) gp > round(R) *** at top-level: round(R) *** ^-------- *** round: precision too low in roundr (precision loss in truncation). *** Break loop: type 'break' to go back to GP prompt break> break (15:54) gp > \p realprecision = 38 significant digits (15:54) gp > \p200 realprecision = 211 significant digits (200 digits displayed) (15:54) gp > R=1 %39 = 1 (15:54) gp > X=polroots(Q) %40 = [-2.1766392753506720687180759370903410741469753767879156590694081422344976242231582592138775308586947915892440324900629748009891632181232888851071750894715310976983902900759617794326271760869517641090737 + 0.E-211*I, -1.4552730944755529730079541428404195057019710619502361817740284165219338137802491735891605672151316245468508278544894507408641377584786580475691593121744249591367686638675120014065176590848680096948920 + 0.E-211*I, -0.81507913996118546183745835334303919268269333304015073306665065321868967588497601407092711374118922465494841558358428061904186811054659229481037025991189055809209463500903914574208653119971695859830885 + 0.E-211*I, 1.6793446099735060222379333362773123397125643731423164950440252377521738590155274982246417878733357978310938763549394729067605917489951526344014967571612632911544270837969181700005392248945930128953711 + 0.E-211*I, 2.7676468998139044813255550969964874328190753986359860788660619742229472548728559486493234239416798429599493995731972332541345773381533865930852079043965833237728265051555947565806921414769437195069035 + 0.E-211*I]~ (15:54) gp > X=real(X) %41 = [-2.1766392753506720687180759370903410741469753767879156590694081422344976242231582592138775308586947915892440324900629748009891632181232888851071750894715310976983902900759617794326271760869517641090737, -1.4552730944755529730079541428404195057019710619502361817740284165219338137802491735891605672151316245468508278544894507408641377584786580475691593121744249591367686638675120014065176590848680096948920, -0.81507913996118546183745835334303919268269333304015073306665065321868967588497601407092711374118922465494841558358428061904186811054659229481037025991189055809209463500903914574208653119971695859830885, 1.6793446099735060222379333362773123397125643731423164950440252377521738590155274982246417878733357978310938763549394729067605917489951526344014967571612632911544270837969181700005392248945930128953711, 2.7676468998139044813255550969964874328190753986359860788660619742229472548728559486493234239416798429599493995731972332541345773381533865930852079043965833237728265051555947565806921414769437195069035]~ (15:54) gp > R=1;forperm(5,s,R*=(y+sum(n=1,5,X[s[n]]*u[n]))) (15:54) gp > R %43 = y^120 + 1.3117531807436602782 E-209*y^119 - 2700.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^118 - 5.240564220381767931 E-205*y^117 + 3515490.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^116 - 8.385999144055449422 E-201*y^115 - 2941059000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^114 - 2.0904355851304953778 E-197*y^113 + 1777297395055.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^112 + 4.353744526553766590 E-194*y^111 - 826923180241800.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^110 + 2.343491545802817572 E-190*y^109 + 308333101945192510.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^108 + 3.435467727395621773 E-187*y^107 - 94694043847528345100.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^106 + 1.9416200280578760467 E-184*y^105 + 24433237971035864924275.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^104 + 2.087885631038543672 E-183*y^103 - 5376233160316134541531400.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^102 - 4.131078842935713435 E-179*y^101 + 1020567781879348259569748624.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^100 - 1.6015639631402892536 E-176*y^99 - 168678245702960424035497929000.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^98 - 1.9543146038097386397 E-174*y^97 + 24453685182038118959187459147320.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^96 + 1.6659475847837840691 E-172*y^95 - 3128400158918188453168932998294000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*y^9[+++] (15:54) gp > round(R) %44 = y^120 - 2700*y^118 + 3515490*y^116 - 2941059000*y^114 + 1777297395055*y^112 - 826923180241800*y^110 + 308333101945192510*y^108 - 94694043847528345100*y^106 + 24433237971035864924275*y^104 - 5376233160316134541531400*y^102 + 1020567781879348259569748624*y^100 - 168678245702960424035497929000*y^98 + 24453685182038118959187459147320*y^96 - 3128400158918188453168932998294000*y^94 + 354940729606632175277410832805437980*y^92 - 35862297516226338578321083235249168800*y^90 + 3237880062041330441841625666774692492565*y^88 - 261978185438492022523229182658164803822900*y^86 + 19040635071508774613919329587728056601358850*y^84 - 1245543822543911134760634367379359782045819800*y^82 + 73449702876006956019459920354856582802224792265*y^80 - 3909561837926691604963011076256852125948669102000*y^78 + 188019432728602589469917906892150606183460437365730*y^76 - 8175816071582320486409110929124462944884258734756500*y^74 + 321607953729666898250556357966290378208495120055205790*y^72 - 11447392587462808219222815584009338857946980294945930900*y^70 + 368719073066756833335547448358391004279173583083203778130*y^68 - 10745599367839809852079685586236931560838130627453144751800*y^66 + 283244938265153298931667264897462478994051007866834953684025*y^64 - 6749240851609020562989302010744238445797249738570310401122800*y^62 + 145273847726386147363500701287690803304586818223940571011198050*y^60 - 2821954401478719220088272140604934150893282592012206449558317500*y^58 + 49413030563043764839681129994624207942190454658912060022376371845*y^56 - 778869835873988795020793218715911396382287387854333934696114033000*y^54 + 11033698687890523199903713340701816384916005514772058999223536852140*y^52 - 140217253009967172725414908223293197880715080607276107811011446276400*y^50 + 1595063347638886437860054683304076154720196484916095434482667520724120*y^48 - 16202862866651499804697854215400466168484828903442356659349786026568200*y^46 + 146567993744881579673661441773236999956396110498669839906868699927839200*y^44 - 1176939959991143783208891488840891361773995924934289912393625009832818800*y^42 + 8359664812748990273286473376875521692655728784330672812116593350394199715*y^40 - 52310262349563344599344230917984980306055469307324911072897551957562988500*y^38 + 287048070568446852869566732564680782442232804556166557400435946179726451230*y^36 - 1374115362251798727533625974075215821001519124782736844064434287724300500000*y^34 + 5704301008451857208171373662351036839519306762805112853810697832262771908255*y^32 - 20394957134234905921556622195184657978867041535433255138673284231711400536800*y^30 + 62310449911440787201053083847144158610667966754755570856605842692566530862210*y^28 - 161191489930362328137220368861486290866197461549754122579412553851901430782100*y^26 + 349309386290587574409986882343609416171819905750309922601728861381109836348625*y^24 - 626106612709560699835746026836161539913303401564424908740765053695957544829800*y^22 + 914167085171154252937069490938587533226039399666012150739751456922139117408496*y^20 - 1067166364021928255903468421147139935541665339390470747222259998278539256003200*y^18 + 973051095812980735173741504509980939220960286025224379110201378113364921301760*y^16 - 67259429238449573328840660[+++] (15:55) gp > R=round(R) %45 = y^120 - 2700*y^118 + 3515490*y^116 - 2941059000*y^114 + 1777297395055*y^112 - 826923180241800*y^110 + 308333101945192510*y^108 - 94694043847528345100*y^106 + 24433237971035864924275*y^104 - 5376233160316134541531400*y^102 + 1020567781879348259569748624*y^100 - 168678245702960424035497929000*y^98 + 24453685182038118959187459147320*y^96 - 3128400158918188453168932998294000*y^94 + 354940729606632175277410832805437980*y^92 - 35862297516226338578321083235249168800*y^90 + 3237880062041330441841625666774692492565*y^88 - 261978185438492022523229182658164803822900*y^86 + 19040635071508774613919329587728056601358850*y^84 - 1245543822543911134760634367379359782045819800*y^82 + 73449702876006956019459920354856582802224792265*y^80 - 3909561837926691604963011076256852125948669102000*y^78 + 188019432728602589469917906892150606183460437365730*y^76 - 8175816071582320486409110929124462944884258734756500*y^74 + 321607953729666898250556357966290378208495120055205790*y^72 - 11447392587462808219222815584009338857946980294945930900*y^70 + 368719073066756833335547448358391004279173583083203778130*y^68 - 10745599367839809852079685586236931560838130627453144751800*y^66 + 283244938265153298931667264897462478994051007866834953684025*y^64 - 6749240851609020562989302010744238445797249738570310401122800*y^62 + 145273847726386147363500701287690803304586818223940571011198050*y^60 - 2821954401478719220088272140604934150893282592012206449558317500*y^58 + 49413030563043764839681129994624207942190454658912060022376371845*y^56 - 778869835873988795020793218715911396382287387854333934696114033000*y^54 + 11033698687890523199903713340701816384916005514772058999223536852140*y^52 - 140217253009967172725414908223293197880715080607276107811011446276400*y^50 + 1595063347638886437860054683304076154720196484916095434482667520724120*y^48 - 16202862866651499804697854215400466168484828903442356659349786026568200*y^46 + 146567993744881579673661441773236999956396110498669839906868699927839200*y^44 - 1176939959991143783208891488840891361773995924934289912393625009832818800*y^42 + 8359664812748990273286473376875521692655728784330672812116593350394199715*y^40 - 52310262349563344599344230917984980306055469307324911072897551957562988500*y^38 + 287048070568446852869566732564680782442232804556166557400435946179726451230*y^36 - 1374115362251798727533625974075215821001519124782736844064434287724300500000*y^34 + 5704301008451857208171373662351036839519306762805112853810697832262771908255*y^32 - 20394957134234905921556622195184657978867041535433255138673284231711400536800*y^30 + 62310449911440787201053083847144158610667966754755570856605842692566530862210*y^28 - 161191489930362328137220368861486290866197461549754122579412553851901430782100*y^26 + 349309386290587574409986882343609416171819905750309922601728861381109836348625*y^24 - 626106612709560699835746026836161539913303401564424908740765053695957544829800*y^22 + 914167085171154252937069490938587533226039399666012150739751456922139117408496*y^20 - 1067166364021928255903468421147139935541665339390470747222259998278539256003200*y^18 + 973051095812980735173741504509980939220960286025224379110201378113364921301760*y^16 - 67259429238449573328840660[+++] (15:55) gp > factor(R) %46 = [y^20 - 450*y^18 + 75637*y^16 - 6024850*y^14 - 731400*y^13 + 246145716*y^12 + 123892800*y^11 - 5401633890*y^10 - 5616865800*y^9 + 61640362773*y^8 + 101051960280*y^7 - 300394517290*y^6 - 704273391600*y^5 + 172925326649*y^4 + 1054138168200*y^3 + 213005704680*y^2 - 413765954400*y - 144539954032 1] [y^20 - 450*y^18 + 75637*y^16 - 6024850*y^14 + 731400*y^13 + 246145716*y^12 - 123892800*y^11 - 5401633890*y^10 + 5616865800*y^9 + 61640362773*y^8 - 101051960280*y^7 - 300394517290*y^6 + 704273391600*y^5 + 172925326649*y^4 - 1054138168200*y^3 + 213005704680*y^2 + 413765954400*y - 144539954032 1] [y^20 - 450*y^18 + 77545*y^16 - 6438250*y^14 + 264789420*y^12 - 5091796950*y^10 + 46982138145*y^8 - 211656814450*y^6 + 436190509025*y^4 - 329322564900*y^2 + 652221392 1] [y^20 - 450*y^18 + 81997*y^16 - 7869250*y^14 - 400680*y^13 + 435764876*y^12 + 96163200*y^11 - 14372051370*y^10 - 7725358440*y^9 + 280244896293*y^8 + 269545964760*y^7 - 3016295033170*y^6 - 4311507714480*y^5 + 14574050227929*y^4 + 27631419121800*y^3 - 13432963723160*y^2 - 37322980166880*y - 6498554091632 1] [y^20 - 450*y^18 + 81997*y^16 - 7869250*y^14 + 400680*y^13 + 435764876*y^12 - 96163200*y^11 - 14372051370*y^10 + 7725358440*y^9 + 280244896293*y^8 - 269545964760*y^7 - 3016295033170*y^6 + 4311507714480*y^5 + 14574050227929*y^4 - 27631419121800*y^3 - 13432963723160*y^2 + 37322980166880*y - 6498554091632 1] [y^20 - 450*y^18 + 85177*y^16 - 8855050*y^14 + 553327356*y^12 - 21320341830*y^10 + 498141157953*y^8 - 6639339540130*y^6 + 44493936998369*y^4 - 116308490978340*y^2 + 13921650463568 1] (15:55) gp > factor(R)[1] *** at top-level: factor(R)[1] *** ^--- *** incorrect type in _[_] OCcompo1 [not a vector] (t_MAT). *** Break loop: type 'break' to go back to GP prompt break> break (15:55) gp > factor(R)[1,1] %47 = y^20 - 450*y^18 + 75637*y^16 - 6024850*y^14 - 731400*y^13 + 246145716*y^12 + 123892800*y^11 - 5401633890*y^10 - 5616865800*y^9 + 61640362773*y^8 + 101051960280*y^7 - 300394517290*y^6 - 704273391600*y^5 + 172925326649*y^4 + 1054138168200*y^3 + 213005704680*y^2 - 413765954400*y - 144539954032 (15:55) gp > R1=factor(R)[1,1] %48 = y^20 - 450*y^18 + 75637*y^16 - 6024850*y^14 - 731400*y^13 + 246145716*y^12 + 123892800*y^11 - 5401633890*y^10 - 5616865800*y^9 + 61640362773*y^8 + 101051960280*y^7 - 300394517290*y^6 - 704273391600*y^5 + 172925326649*y^4 + 1054138168200*y^3 + 213005704680*y^2 - 413765954400*y - 144539954032 (15:56) gp > polgalois(R1) *** at top-level: polgalois(R1) *** ^------------- *** polgalois: sorry, galois of degree higher than 11 is not yet implemented. *** Break loop: type 'break' to go back to GP prompt break> break (15:56) gp > nfsplitting(R1) %49 = y^20 - 450*y^18 + 75637*y^16 - 6024850*y^14 - 731400*y^13 + 246145716*y^12 + 123892800*y^11 - 5401633890*y^10 - 5616865800*y^9 + 61640362773*y^8 + 101051960280*y^7 - 300394517290*y^6 - 704273391600*y^5 + 172925326649*y^4 + 1054138168200*y^3 + 213005704680*y^2 - 413765954400*y - 144539954032 (15:56) gp > nfsplitting(Q) %50 = x^20 - 90*x^18 + 3329*x^16 - 65570*x^14 + 743740*x^12 - 4903454*x^10 + 18122025*x^8 - 35039466*x^6 + 33778225*x^4 - 14887700*x^2 + 2382032 (15:56) gp > Q %51 = x^5 - 9*x^3 - 4*x^2 + 17*x + 12 (15:56) gp > factormod(Q,2) %52 = [ Mod(1, 2)*x 1] [Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2) 2] (15:57) gp > factormod(Q,3) %53 = [ Mod(1, 3)*x 1] [Mod(1, 3)*x^4 + Mod(2, 3)*x + Mod(2, 3) 1] (15:57) gp > factormod(Q,5) %54 = [ Mod(1, 5)*x + Mod(3, 5) 1] [Mod(1, 5)*x^4 + Mod(2, 5)*x^3 + Mod(1, 5)*x + Mod(4, 5) 1] (15:57) gp > poldisc(Q) %55 = 2382032 (15:57) gp > factor(%) %56 = [ 2 4] [53 3] (15:57) gp > factormod(Q,7) %57 = [Mod(1, 7)*x^5 + Mod(5, 7)*x^3 + Mod(3, 7)*x^2 + Mod(3, 7)*x + Mod(5, 7) 1]