tests/rtest_extensions.mac

   1 /* Some cubic polynomials */
   2
   3 block([s,v],s:splitfield(x^3-x-1,x),v:listofvars(s)[1],subst(a,v,s));
   4 [a^6-6*a^4+9*a^2+23,-((a^4-5*a^2+9*a+4)/18),(a^4-5*a^2+4)/9,
   5 -((a^4-5*a^2-9*a+4)/18)]$
   6
   7 /* x^3-x-1  splits completely in Q[a] and one recognizes the roots */
   8
   9 algfac(x^3-x-1,a^6-6*a^4+9*a^2+23);
  10 ((9*x-a^4+5*a^2-4)*(18*x+a^4-5*a^2-9*a+4)*(18*x+a^4-5*a^2+9*a+4))/2916$
  11
  12 /* Adding just one root it factors partially */
  13
  14 algfac(x^3-x-1,a^3-a-1);
  15 (x-a)*(x^2+a*x+a^2-1)$
  16
  17 /* Adding a root of the second factor completes the factorization */
  18
  19 primelmt(x^2+a*x+a^2-1,a^3-a-1,b);
  20 [b^6-6*b^4+9*b^2+23,x+2*a]$
  21
  22 /* One recovers the splitting polynomial obtained by splitfield. Moreover
  23 this is an example where the prime element is of the form x+n*a with n > 1 */
  24
  25 bdiscr(1,x,x^2,x^3-x-1);
  26 ''(rat(-23))$
  27
  28 poly_discriminant(x^3-x-1,x);
  29 -23;
  30
  31 /* If x is a root, it is an algebraic integer, because the equation is monic, and so is
  32 x^2, then since 23 is prime, {1,x,x^2} is a basis of the algebraic integers in Q[x] */
  33
  34 /* Works for a factorized polynomial */
  35
  36 block([s,v],s:splitfield((x^2-3)*(x^2-2),x),v:listofvars(s)[1],subst(a,v,s));
  37 [a^4-10*a^2+1,-((a^3-9*a)/2),(a^3-9*a)/2,(a^3-11*a)/2,-((a^3-11*a)/2)]$
  38
  39 /* One can see that as the successive adjunction of sqrt(2) and sqrt(3),
  40 that is  Q[sqrt(2),sqrt(3)]   */
  41
  42 primelmt(a^2-2,b^2-3,c);
  43 [c^4-10*c^2+1,b+a]$
  44
  45 /* Irreducible quartic polynomials */
  46
  47 algfac(x^4-2,a^4-2);
  48 (x-a)*(x+a)*(x^2+a^2)$
  49
  50 /* This is an example from S. Lang, Algebra. If a=2^(1/4), it generates
  51 a subfield (1,a,a^2,a^3) of dimension 4, which doesn't contain all roots.
  52 One needs to add %i to get all of them. Thus the Galois group has 8 elements
  53 and has subgroups of order 4 and 2 */
  54
  55 primelmt(a^4-2,b^2+1,c);
  56 [c^8+4*c^6+2*c^4+28*c^2+1,b+a]$
  57
  58 algfac(x^4-2,c^8+4*c^6+2*c^4+28*c^2+1);
  59 ((24*x-c^6-5*c^4-13*c^2-29)*(24*x+c^6+5*c^4+13*c^2+29)
  60                                   *(24*x-5*c^7-19*c^5-5*c^3-151*c)
  61                                   *(24*x+5*c^7+19*c^5+5*c^3+151*c))/331776$
  62
  63 /* Another example with Galois group of order 8 */
  64
  65 block([s,v],s:splitfield(x^4+10*x^2-96*x-71,x),v:listofvars(s)[1],subst(a,v,s[1]));
  66 a^8+148*a^6-576*a^5+9814*a^4-42624*a^3+502260*a^2+1109952*a+18860337$
  67
  68 /* With a small change the Galois group is of order 24. The computation takes much
  69 longer */
  70
  71 block([s,v],s:splitfield(x^4+10*x^2-96*x-72,x),v:listofvars(s)[1],subst(a,v,s[1]));
  72 a^24+700*a^22-8640*a^21+202860*a^20-5040000*a^19+85120320*a^18
  73            -1084446720*a^17+28192407600*a^16-307448294400*a^15
  74            +3879455590080*a^14-79002155473920*a^13+862062450860352*a^12
  75            -6341012384716800*a^11+173144130732249600*a^10
  76            -861539021970186240*a^9+6648562403288386560*a^8
  77            -183678612853284864000*a^7+785273136304331653120*a^6
  78            +746838863325592289280*a^5+176781240435133218734080*a^4
  79            +116524085058329916211200*a^3+6983186268545823770542080*a^2
  80            -29426422063945596083896320*a+430620712916420842049765376$
  81
  82 /* For an algebraic curve, such as an hyperbola. Here x^3+2*x*y^2+y^3 is
  83 some algebraic function on the curve. */
  84
  85 algtrace(x^3+2*x*y^2+y^3,y^2-x^2+1,y);
  86 ''(rat(6*x^3-4*x))$
  87
  88 ratsimp(ratsubst(sqrt(-1+x^2),y,x^3+2*x*y^2+y^3)+ratsubst(-sqrt(-1+x^2),y,x^3+2*x*y^2+y^3));
  89 6*x^3-4*x$
  90
  91 /* Taking the trace  with respect to y one projects the curve on the x axis and one gets
  92 an algebraic function of x */
  93
  94 /* A complicated example with a solvable quintic from D.S.Dummit, Mathematics of computation
  95 57,195(1991)387 . This is clearly at the limit of what maxima can do in reasonable time. Dummit
  96 indeed shows the the Galois group is the Frobenius group of order 20. */
  97
  98 block([s,v],s:splitfield(x^5+15*x+12,x),v:listofvars(s)[1],subst(a,v,s));
  99 [a^20+60*a^16-936*a^15+91350*a^12+605880*a^11+1417536*a^10+36463500*a^8
 100             +122488200*a^7+84862080*a^6-91103616*a^5+1032800625*a^4
 101             +1697841000*a^3+7485825600*a^2+4760933760*a+10933303536,
 102         -((131880620547249*a^19+59683129753141*a^18-304524778993866*a^17
 103                               +793203234964096*a^16+4124149330829589*a^15
 104                               -117172981627806003*a^14-71306340933544902*a^13
 105                               +298650865231229952*a^12
 106                               +11251695090720035763*a^11
 107                               +88411847804251841367*a^10
 108                               +196440402174914992122*a^9
 109                               -38341429061015510352*a^8
 110                               +4577341927136007203127*a^7
 111                               +17087648569913092051863*a^6
 112                               +4481766149602673412342*a^5
 113                               -15341521613198393243376*a^4
 114                               +60345466908156421500816*a^3
 115                               +71102697168836172551184*a^2
 116                               +1112228012805629359857696*a
 117                               +1452913156334429802008064)
 118          /948905619130801728000000),
 119         (34941883569106*a^19-19586610011161*a^18-28628104018984*a^17
 120                             +62261757956409*a^16+2067346063311366*a^15
 121                             -33646881858167757*a^14+15872125201726692*a^13
 122                             +31165731682124373*a^12+3118730624434774902*a^11
 123                             +19424644835781405573*a^10
 124                             +34928604068980953168*a^9-38639423702912510133*a^8
 125                             +1269187385301553504698*a^7
 126                             +3667076703510023078577*a^6
 127                             -455062882588288189452*a^5
 128                             -6073679019448320564969*a^4
 129                             +39452666543681343198264*a^3
 130                             +47232807378617176983216*a^2
 131                             +172758700340547215707584*a
 132                             +55253832604158366243216)
 133          /79875691357815936000000,
 134         -((89591685911602852*a^19+2456426401544275*a^18-79829287283577388*a^17
 135                                 -111266747126642685*a^16
 136                                 +6693211742640744216*a^15
 137                                 -85247275218262553049*a^14
 138                                 -7299844963479831120*a^13
 139                                 +74488279542565111191*a^12
 140                                 +8274877365995836629420*a^11
 141                                 +53413178567062212453753*a^10
 142                                 +121592356188979166775396*a^9
 143                                 -50703322221226476748215*a^8
 144                                 +3190417858093889386657296*a^7
 145                                 +11697039806926436578278045*a^6
 146                                 +5565042124477506830126808*a^5
 147                                 -21907925941416463634269203*a^4
 148                                 +111830991479469725159092680*a^3
 149                                 +226384145027764567538132592*a^2
 150                                 +505623402016123027927501440*a
 151                                 +199640907190542149950414896)
 152          /278840229388945773235200000),
 153         (87530181817718523*a^19+1215087725977030782*a^18
 154                                -953928919736667912*a^17
 155                                -1313560049466028133*a^16
 156                                +9462158957065791483*a^15
 157                                -30528725605743381606*a^14
 158                                -1150352116504017502104*a^13
 159                                +734949752502164464839*a^12
 160                                +9542188140971754165201*a^11
 161                                +159271468071468573427074*a^10
 162                                +789030615124104571528344*a^9
 163                                +1002612772932828989646321*a^8
 164                                +1441683814063122556892289*a^7
 165                                +53856342854315366109409926*a^6
 166                                +116471037468993801002554824*a^5
 167                                -71683288419330233672409027*a^4
 168                                -112235865378225651840900168*a^3
 169                                +1132885879666976209353768288*a^2
 170                                +1236791680392483380748206592*a
 171                                +5963360303581465557565858608)
 172          /1917026577049002190992000000,
 173         -((87530181817718523*a^19+1215087725977030782*a^18
 174                                 -953928919736667912*a^17
 175                                 -1313560049466028133*a^16
 176                                 +9462158957065791483*a^15
 177                                 -30528725605743381606*a^14
 178                                 -1150352116504017502104*a^13
 179                                 +734949752502164464839*a^12
 180                                 +9542188140971754165201*a^11
 181                                 +159271468071468573427074*a^10
 182                                 +789030615124104571528344*a^9
 183                                 +1002612772932828989646321*a^8
 184                                 +1441683814063122556892289*a^7
 185                                 +53856342854315366109409926*a^6
 186                                 +116471037468993801002554824*a^5
 187                                 -71683288419330233672409027*a^4
 188                                 -112235865378225651840900168*a^3
 189                                 +1132885879666976209353768288*a^2
 190                                 -680234896656518810243793408*a
 191                                 +5963360303581465557565858608)
 192          /3834053154098004381984000000)]$
 193
 194 /* Strangely this slightly different example of Dummit has a Galois group of order 10
 195 and is obtained much faster. */
 196
 197 block([s,v],s:splitfield(x^5-5*x+12,x),v:listofvars(s)[1],subst(a,v,s));
 198 [a^10-20*a^8-60*a^7+230*a^6+612*a^5-400*a^4-4020*a^3+39865*a^2-167220*a
 199             +196036,
 200         -((4402021*a^9+8967410*a^8-71413351*a^7-408236968*a^6+199818827*a^5
 201                      +3092804186*a^4+4100015915*a^3-6578347876*a^2
 202                      +173297643052*a-409949875488)
 203          /34533412800),
 204         (129271*a^9-1030711*a^8-6736419*a^7+9580539*a^6+215434401*a^5
 205                    +342066891*a^4-1374973581*a^3-3638935299*a^2+647047544*a
 206                    -49872596204)
 207          /34533412800,
 208         (10382*a^9+27879*a^8-140650*a^7-1021019*a^6-626422*a^5+5112817*a^4
 209                   +14638214*a^3+12817495*a^2+373402956*a-783084692)
 210          /178929600,
 211         (1134512*a^9+2308737*a^8-18765741*a^7-110380420*a^6+52641936*a^5
 212                     +881981807*a^4+1324907097*a^3-2706594556*a^2+41658559300*a
 213                     -104470966864)
 214          /8633353200,
 215         -((1134512*a^9+2308737*a^8-18765741*a^7-110380420*a^6+52641936*a^5
 216                      +881981807*a^4+1324907097*a^3-2706594556*a^2
 217                      +33025206100*a-104470966864)
 218                    /17266706400)]$
 219
 220 /* And this one is even more degenerate */
 221
 222 block([s,v],s:splitfield(x^5-110*x^3-55*x^2+2310*x+979,x),v:listofvars(s)[1],subst(a,v,s));
 223
 224 [a^5-110*a^3-55*a^2+2310*a+979,(a^3-3*a^2-72*a+99)/25,
 225  (a^4-4*a^3-94*a^2+196*a+1276)/125,-((a^4+a^3-84*a^2-89*a+671)/125),
 226  (a^2-2*a-44)/5,a];
 227
 228 /* Since splitfield extensively uses algnorm and factorization in extensions, one can
 229 assume that they work correctly */