Logging to /Users/s/tmp/pari-11.09 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 (16:11) gp > cos(2*Pi/5) %1 = 0.30901699437494742410229341718281905886 (16:11) gp > (sqrt(5)-1)/4 %2 = 0.30901699437494742410229341718281905886 (16:11) gp > z=Mod(x,polcyclo(7)) %3 = Mod(x, x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (16:19) gp > z+1/z %4 = Mod(-x^5 - x^4 - x^3 - x^2 - 1, x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (16:19) gp > prod(n=1,3,y-z^n-z^-n) %5 = Mod(y^3 + y^2 - 2*y - 1, x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (16:20) gp > lift(%) %6 = y^3 + y^2 - 2*y - 1 (16:20) gp > poldisc(%) %7 = 49 (16:20) gp > polroots(%6) %8 = [-1.8019377358048382524722046390148901023 + 0.E-38*I, -0.44504186791262880857780512899358951893 + 0.E-38*I, 1.2469796037174670610500097680084796213 + 0.E-38*I]~ (16:21) gp > polgalois(%6) %9 = [3, 1, 1, "A3"] (16:21) gp > %8[1] %10 = -1.8019377358048382524722046390148901023 + 0.E-38*I (16:23) gp > lindep([%8[1],sqrt(3),1]) %11 = [-9773888376, -9224651701, -1634372862]~ (16:23) gp > lindep([%8[1],sqrt(3)*7^(1/3),1]) %12 = [-4868353050, -6578354068, 13023551991]~ (16:24) gp > lindep([%8[1],sqrt(3)*7^(1/3),7^(1/3)]) %13 = [-2308266673, -4521759023, 5657581577]~ (16:24) gp > ?poltschirnhaus poltschirnhaus(x): random Tschirnhausen transformation of the polynomial x. (16:24) gp > P=%6 %14 = y^3 + y^2 - 2*y - 1 (16:24) gp > ??poltschirnhaus poltschirnhaus(x): Applies a random Tschirnhausen transformation to the polynomial x, which is assumed to be nonconstant and separable, so as to obtain a new equation for the 'etale algebra defined by x. This is for instance useful when computing resolvents, hence is used by the polgalois function. The library syntax is GEN tschirnhaus(GEN x). (16:24) gp > subst(P,y,y-1/3) %15 = y^3 - 7/3*y - 7/27 (16:25) gp > Q=% %16 = y^3 - 7/3*y - 7/27 (16:25) gp > subst(Q,y,(a*y+b)/(1+c*y)) %17 = ((27*a^3 - 63*c^2*a - 7*c^3)*y^3 + (81*b*a^2 - 126*c*a + (-63*c^2*b - 21*c^2))*y^2 + ((81*b^2 - 63)*a + (-126*c*b - 21*c))*y + (27*b^3 - 63*b - 7))/(27*c^3*y^3 + 81*c^2*y^2 + 81*c*y + 27) (16:26) gp > numerator(%) %18 = (27*a^3 - 63*c^2*a - 7*c^3)*y^3 + (81*b*a^2 - 126*c*a + (-63*c^2*b - 21*c^2))*y^2 + ((81*b^2 - 63)*a + (-126*c*b - 21*c))*y + (27*b^3 - 63*b - 7) (16:26) gp > subst(Q,y,y/(1+c*y)) %19 = ((-7*c^3 - 63*c^2 + 27)*y^3 + (-21*c^2 - 126*c)*y^2 + (-21*c - 63)*y - 7)/(27*c^3*y^3 + 81*c^2*y^2 + 81*c*y + 27) (16:26) gp > numerator(%) %20 = (-7*c^3 - 63*c^2 + 27)*y^3 + (-21*c^2 - 126*c)*y^2 + (-21*c - 63)*y - 7 (16:26) gp > subst(%,c,-3) %21 = -351*y^3 + 189*y^2 - 7 (16:27) gp > subst(Q,y,(y+b)/(1+c*y)) %22 = ((-7*c^3 - 63*c^2 + 27)*y^3 + ((-63*c^2 + 81)*b + (-21*c^2 - 126*c))*y^2 + (81*b^2 - 126*c*b + (-21*c - 63))*y + (27*b^3 - 63*b - 7))/(27*c^3*y^3 + 81*c^2*y^2 + 81*c*y + 27) (16:27) gp > numerator(%) %23 = (-7*c^3 - 63*c^2 + 27)*y^3 + ((-63*c^2 + 81)*b + (-21*c^2 - 126*c))*y^2 + (81*b^2 - 126*c*b + (-21*c - 63))*y + (27*b^3 - 63*b - 7) (16:27) gp > poldisc(x^3-a) %24 = -27*a^2 (16:28) gp > poldisc(P) %25 = 49 (16:28) gp > nfinit(P) %26 = [y^3 + y^2 - 2*y - 1, [3, 0], 49, 1, [[1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [16, -29, 20; 16, -7, -29; 16, 20, -7], [3, -1, -1; -1, 5, -2; -1, -2, 5], [7, 0, 6; 0, 7, 3; 0, 0, 1], [3, 1, 1; 1, 2, 1; 1, 1, 2], [7, [-2, 2, -1; 1, -2, 0; 0, 1, -3]], [7]], [-1.8019377358048382524722046390148901023, -0.44504186791262880857780512899358951893, 1.2469796037174670610500097680084796213], [1, y, y^2 - 2], [1, 0, 2; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 2, -1, 0, -1, 1; 0, 1, 0, 1, 0, 0, 0, 0, -1; 0, 0, 1, 0, 1, -1, 1, -1, -1]] (16:32) gp > nfinit(P).disc %27 = 49 (16:32) gp > nfinit(x^3+49) %28 = [x^3 + 49, [1, 1], -1323, 7, [[1, 1.9129311827723891011991168395487602829, -3.6593057100229715172380733101194082635; 1, -0.95646559138619455059955841977438014143 + 1.6566469999723020770048742452831871490*I, 1.8296528550114857586190366550597041317 + 3.1690517050933458667388088508315726620*I], [1, 1.9129311827723891011991168395487602829, -3.6593057100229715172380733101194082635; 1, 0.70018140858610752640531582550880700754, 4.9987045601048316253578455058912767937; 1, -2.6131125913584966276044326650575672904, -1.3393988500818601081197721957718685302], [16, 31, -59; 16, 11, 80; 16, -42, -21], [3, 0, 0; 0, 0, -21; 0, -21, 0], [21, 0, 0; 0, 21, 0; 0, 0, 3], [7, 0, 0; 0, 0, -1; 0, -1, 0], [7, [0, 0, -7; 1, 0, 0; 0, -1, 0]], [3, 7]], [-3.6593057100229715172380733101194082635, 1.8296528550114857586190366550597041317 + 3.1690517050933458667388088508315726620*I], [7, x^2, 7*x], [1, 0, 0; 0, 0, 7; 0, 1, 0], [1, 0, 0, 0, 0, -7, 0, -7, 0; 0, 1, 0, 1, 0, 0, 0, 0, 7; 0, 0, 1, 0, -1, 0, 1, 0, 0]] (16:32) gp > %.disc %29 = -1323 (16:33) gp > factor(%) %30 = [-1 1] [ 3 3] [ 7 2] (16:33) gp > nfsplitting(polcyclo(7)) %31 = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 (16:34) gp > nfinit(polcyclo(7)) %32 = [x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, [0, 3], -16807, 1, [[1, -0.90096886790241912623610231950744505117 + 0.43388373911755812047576833284835875461*I, 0.62348980185873353052500488400423981063 - 0.78183148246802980870844452667405775023*I, -0.22252093395631440428890256449679475947 + 0.97492791218182360701813168299393121723*I, -0.22252093395631440428890256449679475947 - 0.97492791218182360701813168299393121723*I, 0.62348980185873353052500488400423981063 + 0.78183148246802980870844452667405775023*I; 1, 0.62348980185873353052500488400423981063 + 0.78183148246802980870844452667405775023*I, -0.22252093395631440428890256449679475947 + 0.97492791218182360701813168299393121723*I, -0.90096886790241912623610231950744505117 + 0.43388373911755812047576833284835875461*I, -0.90096886790241912623610231950744505117 - 0.43388373911755812047576833284835875461*I, -0.22252093395631440428890256449679475947 - 0.97492791218182360701813168299393121723*I; 1, -0.22252093395631440428890256449679475947 + 0.97492791218182360701813168299393121723*I, -0.90096886790241912623610231950744505117 - 0.43388373911755812047576833284835875461*I, 0.62348980185873353052500488400423981063 - 0.78183148246802980870844452667405775023*I, 0.62348980185873353052500488400423981063 + 0.78183148246802980870844452667405775023*I, -0.90096886790241912623610231950744505117 + 0.43388373911755812047576833284835875461*I], [1, -0.46708512878486100576033398665908629656, -0.15834168060929627818343964266981793960, 0.75240697822550920272922911849713645777, -1.1974488461381380113070342474907259767, 1.4053212843267633392334494106782975609; 1, -1.3348526070199772467118706523558038058, 1.4053212843267633392334494106782975609, -1.1974488461381380113070342474907259767, 0.75240697822550920272922911849713645777, -0.15834168060929627818343964266981793960; 1, 1.4053212843267633392334494106782975609, 0.75240697822550920272922911849713645777, -0.46708512878486100576033398665908629656, -1.3348526070199772467118706523558038058, -1.1974488461381380113070342474907259767; 1, -0.15834168060929627818343964266981793960, -1.1974488461381380113070342474907259767, -1.3348526070199772467118706523558038058, -0.46708512878486100576033398665908629656, 0.75240697822550920272922911849713645777; 1, 0.75240697822550920272922911849713645777, -1.3348526070199772467118706523558038058, -0.15834168060929627818343964266981793960, 1.4053212843267633392334494106782975609, -0.46708512878486100576033398665908629656; 1, -1.1974488461381380113070342474907259767, -0.46708512878486100576033398665908629656, 1.4053212843267633392334494106782975609, -0.15834168060929627818343964266981793960, -1.3348526070199772467118706523558038058], [16, -7, -3, 12, -19, 22; 16, -21, 22, -19, 12, -3; 16, 22, 12, -7, -21, -19; 16, -3, -19, -21, -7, 12; 16, 12, -21, -3, 22, -7; 16, -19, -7, 22, -3, -21], [6, -1, -1, -1, -1, -1; -1, -1, -1, -1, -1, -1; -1, -1, -1, -1, -1, 6; -1, -1, -1, -1, 6, -1; -1, -1, -1, 6, -1, -1; -1, -1, 6, -1, -1, -1], [7, 0, 0, 0, 0, 6; 0, 7, 0, 0, 0, 5; 0, 0, 7, 0, 0, 4; 0, 0, 0, 7, 0, 3; 0, 0, 0, 0, 7, 2; 0, 0, 0, 0, 0, 1], [1, -1, 0, 0, 0, 0; -1, -2, -1, -1, -1, -1; 0, -1, 0, 0, 0, 1; 0, -1, 0, 0, 1, 0; 0, -1, 0, 1, 0, 0; 0, -1, 1, 0, 0, 0], [7, [-1, 0, 0, 0, 0, -1; 1, -1, 0, 0, 0, -1; 0, 1, -1, 0, 0, -1; 0, 0, 1, -1, 0, -1; 0, 0, 0, 1, -1, -1; 0, 0, 0, 0, 1, -2]], [7]], [-0.90096886790241912623610231950744505116 + 0.43388373911755812047576833284835875461*I, 0.62348980185873353052500488400423981063 + 0.78183148246802980870844452667405775023*I, -0.22252093395631440428890256449679475947 + 0.97492791218182360701813168299393121723*I], [1, x, x^2, x^3, x^4, x^5], [1, 0, 0, 0, 0, 0; 0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0; 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0; 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0; 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1; 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0]] (16:34) gp > nfsubfields(polcyclo(7)) %33 = [[x + 1, -1], [x^2 + x + 2, x^4 + x^2 + x], [x^3 + x^2 - 2*x - 1, -x^5 - x^4 - x^3 - x^2 - 1], [x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x]] (16:34) gp > polcyclo(21) %34 = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 (16:35) gp > nfsubfields(polcyclo(21)) %35 = [[x - 1, 1], [x^2 - x + 1, -x^7], [x^2 - x + 2, x^11 - x^9 + x^8 + x^4 + x], [x^2 - x - 5, -x^11 - x^9 - x^8 + x^7 + x^4 - 2*x^2 + x + 1], [x^3 - x^2 - 2*x + 1, x^8 - x^6 + x], [x^4 - x^3 - x^2 - 2*x + 4, -x^9 + x^4 - x^2 + x], [x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1, x^11 - x^10 + x^8 - x^7 - x^6 + x^5 - x^3 + x^2 + x - 1], [x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, x^8 + x], [x^6 - x^5 - 6*x^4 + 6*x^3 + 8*x^2 - 8*x + 1, -x^11 + x^10 - x^8 + x^7 - x^5 + x^3 - x^2 + x + 1], [x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1, x]] (16:35) gp > #% %36 = 10 (16:35) gp > nfdisc(polcyclo(21)) %37 = 205924456521 (16:37) gp > factor(%) %38 = [3 6] [7 10] (16:37) gp > nfdiscfactors(polcyclo(21)) %39 = [205924456521, [3, 6; 7, 10]] (16:37) gp > ?nfgaloisapply nfgaloisapply(nf,aut,x): apply the Galois automorphism aut to the object x (element or ideal) in the number field nf. (16:37) gp > ?nfgaloisconj nfgaloisconj(nf,{flag=0},{d}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for details). nf can be simply a polynomial. (16:37) gp > ??nfgaloisapply nfgaloisapply(nf,aut,x): Let nf be a number field as output by nfinit, and let aut be a Galois automorphism of nf expressed by its image on the field generator (such automorphisms can be found using nfgaloisconj). The function computes the action of the automorphism aut on the object x in the number field; x can be a number field element, or an ideal (possibly extended). Because of possible confusion with elements and ideals, other vector or matrix arguments are forbidden. ? nf = nfinit(x^2+1); ? L = nfgaloisconj(nf) %2 = [-x, x]~ ? aut = L[1]; /* the nontrivial automorphism */ ? nfgaloisapply(nf, aut, x) %4 = Mod(-x, x^2 + 1) ? P = idealprimedec(nf,5); /* prime ideals above 5 */ ? nfgaloisapply(nf, aut, P[2]) == P[1] %6 = 0 \\ !!!! ? idealval(nf, nfgaloisapply(nf, aut, P[2]), P[1]) %7 = 1 The surprising failure of the equality test (%7) is due to the fact that although the corresponding prime ideals are equal, their representations are not. (A prime ideal is specified by a uniformizer, and there is no guarantee that applying automorphisms yields the same elements as a direct idealprimedec call.) The automorphism can also be given as a column vector, representing the image of Mod(x, nf.pol) as an algebraic number. This last representation is more efficient and should be preferred if a given automorphism must be used in many such calls. ? nf = nfinit(x^3 - 37*x^2 + 74*x - 37); ? aut = nfgaloisconj(nf)[2]; \\ an automorphism in basistoalg form %2 = -31/11*x^2 + 1109/11*x - 925/11 ? AUT = nfalgtobasis(nf, aut); \\ same in algtobasis form %3 = [16, -6, 5]~ ? v = [1, 2, 3]~; nfgaloisapply(nf, aut, v) == nfgaloisapply(nf, AUT, v) %4 = 1 \\ same result... ? for (i=1,10^5, nfgaloisapply(nf, aut, v)) time = 463 ms. ? for (i=1,10^5, nfgaloisapply(nf, AUT, v)) time = 343 ms. \\ but the latter is faster The library syntax is GEN galoisapply(GEN nf, GEN aut, GEN x). (16:38) gp > factor(x^3-37*x^2+74*x-37) %40 = [x^3 - 37*x^2 + 74*x - 37 1] (16:43) gp > poldisc(x^3-37*x^2+74*x-37) %41 = 165649 (16:43) gp > factor(%0 *** syntax error, unexpected $end, expecting )-> or ',' or ')': factor(%0 *** ^- (16:43) gp > factor(%0) %42 = [11 2] [37 2] (16:43) gp > polgalois(x^3-37*x^2+74*x-37) %43 = [3, 1, 1, "A3"] (16:43) gp > nf=nfinit(x^3-37*x^2+74*x-37) %44 = [x^3 - 37*x^2 + 74*x - 37, [3, 0], 1369, 11, [[1, -1.1576115578454257614823213201242253706, -2.1871008076064092016833700987227610994; 1, -2.1871008076064092016833700987227610994, 4.3447123654518349631656914188469864700; 1, 4.3447123654518349631656914188469864700, -1.1576115578454257614823213201242253706], [1, -1.1576115578454257614823213201242253706, -2.1871008076064092016833700987227610994; 1, -2.1871008076064092016833700987227610994, 4.3447123654518349631656914188469864700; 1, 4.3447123654518349631656914188469864700, -1.1576115578454257614823213201242253706], [16, -19, -35; 16, -35, 70; 16, 70, -19], [3, 1, 1; 1, 25, -12; 1, -12, 25], [37, 0, 3; 0, 37, 27; 0, 0, 1], [13, -1, -1; -1, 2, 1; -1, 1, 2], [37, [12, 7, -3; 1, 15, -1; 0, 1, 10]], [37]], [0.86722984532103622942270198053188667701, 1.2221075198133797530654708265104198738, 34.910662634865584017511827192957693449], [11, x^2 - 34*x + 16, -6*x^2 + 215*x - 206], [1, 10, 324; 0, 6, 215; 0, 1, 34], [1, 0, 0, 0, 7, -3, 0, -3, 8; 0, 1, 0, 1, 3, -1, 0, -1, -1; 0, 0, 1, 0, 1, -2, 1, -2, 2]] (16:43) gp > nfgaloisconj(nf) %45 = [x, -31/11*x^2 + 1109/11*x - 925/11, 31/11*x^2 - 1120/11*x + 1332/11]~ (16:43) gp > aut=nfgaloisconj(nf)[2] %46 = -31/11*x^2 + 1109/11*x - 925/11 (16:44) gp > ?nfalgtobasis nfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zk. (16:44) gp > ??nfalgtobasis nfalgtobasis(nf,x): Given an algebraic number x in the number field nf, transforms it to a column vector on the integral basis nf.zk. ? nf = nfinit(y^2 + 4); ? nf.zk %2 = [1, 1/2*y] ? nfalgtobasis(nf, [1,1]~) %3 = [1, 1]~ ? nfalgtobasis(nf, y) %4 = [0, 2]~ ? nfalgtobasis(nf, Mod(y, y^2+4)) %5 = [0, 2]~ This is the inverse function of nfbasistoalg. The library syntax is GEN algtobasis(GEN nf, GEN x). (16:44) gp > nfgaloisapply(nf,aut,x) %47 = Mod(-31/11*x^2 + 1109/11*x - 925/11, x^3 - 37*x^2 + 74*x - 37) (16:46) gp > nfgaloisapply(nf,aut,x-1) %48 = Mod(-31/11*x^2 + 1109/11*x - 936/11, x^3 - 37*x^2 + 74*x - 37) (16:46) gp > nfgaloisapply(nf,aut,(x-1)/(x^2+2)) *** at top-level: nfgaloisapply(nf,aut,(x-1)/(x^2+2)) *** ^----------------------------------- *** nfgaloisapply: incorrect type in galoisapply (t_RFRAC). *** Break loop: type 'break' to go back to GP prompt break> break (16:47) gp > nfgaloisapply(nf,aut,Mod((x-1)/(x^2+2),nf.pol)) %49 = Mod(34/11737*x^2 - 1255/11737*x + 1809/11737, x^3 - 37*x^2 + 74*x - 37) (16:47) gp > nf.pol %50 = x^3 - 37*x^2 + 74*x - 37 (16:47) gp > P %51 = y^3 + y^2 - 2*y - 1 (16:47) gp > nfinit(P) %52 = [y^3 + y^2 - 2*y - 1, [3, 0], 49, 1, [[1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [16, -29, 20; 16, -7, -29; 16, 20, -7], [3, -1, -1; -1, 5, -2; -1, -2, 5], [7, 0, 6; 0, 7, 3; 0, 0, 1], [3, 1, 1; 1, 2, 1; 1, 1, 2], [7, [-2, 2, -1; 1, -2, 0; 0, 1, -3]], [7]], [-1.8019377358048382524722046390148901023, -0.44504186791262880857780512899358951893, 1.2469796037174670610500097680084796213], [1, y, y^2 - 2], [1, 0, 2; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 2, -1, 0, -1, 1; 0, 1, 0, 1, 0, 0, 0, 0, -1; 0, 0, 1, 0, 1, -1, 1, -1, -1]] (16:47) gp > Corpo=nfinit(P) %53 = [y^3 + y^2 - 2*y - 1, [3, 0], 49, 1, [[1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [1, -1.8019377358048382524722046390148901023, 1.2469796037174670610500097680084796213; 1, -0.44504186791262880857780512899358951893, -1.8019377358048382524722046390148901023; 1, 1.2469796037174670610500097680084796213, -0.44504186791262880857780512899358951893], [16, -29, 20; 16, -7, -29; 16, 20, -7], [3, -1, -1; -1, 5, -2; -1, -2, 5], [7, 0, 6; 0, 7, 3; 0, 0, 1], [3, 1, 1; 1, 2, 1; 1, 1, 2], [7, [-2, 2, -1; 1, -2, 0; 0, 1, -3]], [7]], [-1.8019377358048382524722046390148901023, -0.44504186791262880857780512899358951893, 1.2469796037174670610500097680084796213], [1, y, y^2 - 2], [1, 0, 2; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 2, -1, 0, -1, 1; 0, 1, 0, 1, 0, 0, 0, 0, -1; 0, 0, 1, 0, 1, -1, 1, -1, -1]] (16:47) gp > nfgaloisconj(Corpo) %54 = [y, -y^2 - y + 1, y^2 - 2]~ (16:47) gp > subst(P,y,(1-y-y^2)) %55 = -y^6 - 3*y^5 + y^4 + 7*y^3 + y^2 - 3*y - 1 (16:48) gp > factor(%) %56 = [y^3 + y^2 - 2*y - 1 1] [y^3 + 2*y^2 - y - 1 1] (16:48) gp > P %57 = y^3 + y^2 - 2*y - 1 (16:48) gp > "C(a)[x]/(x^5-x-a) como extensão de C(a)" %58 = "C(a)[x]/(x^5-x-a) como extensão de C(a)" (16:55) gp > poldisc(x^5-x-a) %59 = 3125*a^4 - 256 (16:56) gp > factor(256/3125) %60 = [2 8] [5 -5] (16:57) gp > polgalois(x^5-x-1) %61 = [120, -1, 1, "S5"] (16:59) gp > polgalois(x^5-x-2) %62 = [120, -1, 1, "S5"] (16:59) gp > polgalois(Mod(x^5-x-1,5)) *** at top-level: polgalois(Mod(x^5-x-1,5)) *** ^------------------------- *** polgalois: incorrect type in galois [not in Z[X]] (t_POL). *** Break loop: type 'break' to go back to GP prompt break> break (17:00) gp > factor(x^5-x-1,1) %63 = [x^5 - x - 1 1] (17:00) gp > factor(x^5-x-1,2) %64 = [x^5 - x - 1 1] (17:00) gp > factor(x^5-x-1,3) %65 = [x^5 - x - 1 1] (17:00) gp > factor(Mod(x^5-x-1,3)) %66 = [Mod(1, 3)*x^5 + Mod(2, 3)*x + Mod(2, 3) 1] (17:00) gp > factor(Mod(x^5-x-1,5)) %67 = [Mod(1, 5)*x^5 + Mod(4, 5)*x + Mod(4, 5) 1] (17:00) gp > factor(Mod(x^5-x-1,7)) %68 = [ 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] (17:00) gp > factor(Mod(x^5-x-1,11)) %69 = [Mod(1, 11)*x^5 + Mod(10, 11)*x + Mod(10, 11) 1] (17:00) gp > factor(Mod(x^5-x-1,3)) %70 = [Mod(1, 3)*x^5 + Mod(2, 3)*x + Mod(2, 3) 1] (17:00) gp > factor(Mod(x^5-x-1,2)) %71 = [ 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] (17:00) gp > factor(Mod(x^5-x-1,13)) %72 = [Mod(1, 13)*x^5 + Mod(12, 13)*x + Mod(12, 13) 1] (17:01) gp > factor(Mod(x^5-x-1,17)) %73 = [ 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] (17:01) gp > factor(Mod(x^5-x-1,19)) %74 = [ Mod(1, 19)*x + Mod(6, 19) 2] [Mod(1, 19)*x^3 + Mod(7, 19)*x^2 + Mod(13, 19)*x + Mod(10, 19) 1] (17:01) gp > factor(Mod(x^5-x-1,23)) %75 = [ 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] (17:01) gp > factor(Mod(x^5-x-1,29)) %76 = [ Mod(1, 29)*x + Mod(27, 29) 1] [Mod(1, 29)*x^4 + Mod(2, 29)*x^3 + Mod(4, 29)*x^2 + Mod(8, 29)*x + Mod(15, 29) 1] (17:01) gp > factor(Mod(x^5-x-1,31)) %77 = [ Mod(1, 31)*x + Mod(2, 31) 1] [Mod(1, 31)*x^4 + Mod(29, 31)*x^3 + Mod(4, 31)*x^2 + Mod(23, 31)*x + Mod(15, 31) 1] (17:01) gp > factor(Mod(x^5-x-1,37)) %78 = [ Mod(1, 37)*x^2 + Mod(16, 37)*x + Mod(29, 37) 1] [Mod(1, 37)*x^3 + Mod(21, 37)*x^2 + Mod(5, 37)*x + Mod(14, 37) 1] (17:01) gp > factor(Mod(x^5-x-1,31)) %79 = [ Mod(1, 31)*x + Mod(2, 31) 1] [Mod(1, 31)*x^4 + Mod(29, 31)*x^3 + Mod(4, 31)*x^2 + Mod(23, 31)*x + Mod(15, 31) 1] (17:01) gp > factor(Mod(x^5-x-1,41)) %80 = [ Mod(1, 41)*x + Mod(15, 41) 1] [ Mod(1, 41)*x + Mod(33, 41) 1] [Mod(1, 41)*x^3 + Mod(34, 41)*x^2 + Mod(5, 41)*x + Mod(27, 41) 1] (17:01) gp > factor(Mod(x^5-x-1,71)) %81 = [ Mod(1, 71)*x + Mod(42, 71) 1] [ Mod(1, 71)*x^2 + Mod(43, 71)*x + Mod(8, 71) 1] [Mod(1, 71)*x^2 + Mod(57, 71)*x + Mod(15, 71) 1]