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/* Some cubic polynomials */
block([s,v],s:splitfield(x^3-x-1,x),v:listofvars(s)[1],subst(a,v,s));
[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,
-((a^4-5*a^2-9*a+4)/18)]$
/* x^3-x-1 splits completely in Q[a] and one recognizes the roots */
algfac(x^3-x-1,a^6-6*a^4+9*a^2+23);
((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$
/* Adding just one root it factors partially */
algfac(x^3-x-1,a^3-a-1);
(x-a)*(x^2+a*x+a^2-1)$
/* Adding a root of the second factor completes the factorization */
primelmt(x^2+a*x+a^2-1,a^3-a-1,b);
[b^6-6*b^4+9*b^2+23,x+2*a]$
/* One recovers the splitting polynomial obtained by splitfield. Moreover
this is an example where the prime element is of the form x+n*a with n > 1 */
bdiscr(1,x,x^2,x^3-x-1);
''(rat(-23))$
poly_discriminant(x^3-x-1,x);
-23;
/* If x is a root, it is an algebraic integer, because the equation is monic, and so is
x^2, then since 23 is prime, {1,x,x^2} is a basis of the algebraic integers in Q[x] */
/* Works for a factorized polynomial */
block([s,v],s:splitfield((x^2-3)*(x^2-2),x),v:listofvars(s)[1],subst(a,v,s));
[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)]$
/* One can see that as the successive adjunction of sqrt(2) and sqrt(3),
that is Q[sqrt(2),sqrt(3)] */
primelmt(a^2-2,b^2-3,c);
[c^4-10*c^2+1,b+a]$
/* Irreducible quartic polynomials */
algfac(x^4-2,a^4-2);
(x-a)*(x+a)*(x^2+a^2)$
/* This is an example from S. Lang, Algebra. If a=2^(1/4), it generates
a subfield (1,a,a^2,a^3) of dimension 4, which doesn't contain all roots.
One needs to add %i to get all of them. Thus the Galois group has 8 elements
and has subgroups of order 4 and 2 */
primelmt(a^4-2,b^2+1,c);
[c^8+4*c^6+2*c^4+28*c^2+1,b+a]$
algfac(x^4-2,c^8+4*c^6+2*c^4+28*c^2+1);
((24*x-c^6-5*c^4-13*c^2-29)*(24*x+c^6+5*c^4+13*c^2+29)
*(24*x-5*c^7-19*c^5-5*c^3-151*c)
*(24*x+5*c^7+19*c^5+5*c^3+151*c))/331776$
/* Another example with Galois group of order 8 */
block([s,v],s:splitfield(x^4+10*x^2-96*x-71,x),v:listofvars(s)[1],subst(a,v,s[1]));
a^8+148*a^6-576*a^5+9814*a^4-42624*a^3+502260*a^2+1109952*a+18860337$
/* With a small change the Galois group is of order 24. The computation takes much
longer */
block([s,v],s:splitfield(x^4+10*x^2-96*x-72,x),v:listofvars(s)[1],subst(a,v,s[1]));
a^24+700*a^22-8640*a^21+202860*a^20-5040000*a^19+85120320*a^18
-1084446720*a^17+28192407600*a^16-307448294400*a^15
+3879455590080*a^14-79002155473920*a^13+862062450860352*a^12
-6341012384716800*a^11+173144130732249600*a^10
-861539021970186240*a^9+6648562403288386560*a^8
-183678612853284864000*a^7+785273136304331653120*a^6
+746838863325592289280*a^5+176781240435133218734080*a^4
+116524085058329916211200*a^3+6983186268545823770542080*a^2
-29426422063945596083896320*a+430620712916420842049765376$
/* For an algebraic curve, such as an hyperbola. Here x^3+2*x*y^2+y^3 is
some algebraic function on the curve. */
algtrace(x^3+2*x*y^2+y^3,y^2-x^2+1,y);
''(rat(6*x^3-4*x))$
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));
6*x^3-4*x$
/* Taking the trace with respect to y one projects the curve on the x axis and one gets
an algebraic function of x */
/* A complicated example with a solvable quintic from D.S.Dummit, Mathematics of computation
57,195(1991)387 . This is clearly at the limit of what maxima can do in reasonable time. Dummit
indeed shows the the Galois group is the Frobenius group of order 20. */
block([s,v],s:splitfield(x^5+15*x+12,x),v:listofvars(s)[1],subst(a,v,s));
[a^20+60*a^16-936*a^15+91350*a^12+605880*a^11+1417536*a^10+36463500*a^8
+122488200*a^7+84862080*a^6-91103616*a^5+1032800625*a^4
+1697841000*a^3+7485825600*a^2+4760933760*a+10933303536,
-((131880620547249*a^19+59683129753141*a^18-304524778993866*a^17
+793203234964096*a^16+4124149330829589*a^15
-117172981627806003*a^14-71306340933544902*a^13
+298650865231229952*a^12
+11251695090720035763*a^11
+88411847804251841367*a^10
+196440402174914992122*a^9
-38341429061015510352*a^8
+4577341927136007203127*a^7
+17087648569913092051863*a^6
+4481766149602673412342*a^5
-15341521613198393243376*a^4
+60345466908156421500816*a^3
+71102697168836172551184*a^2
+1112228012805629359857696*a
+1452913156334429802008064)
/948905619130801728000000),
(34941883569106*a^19-19586610011161*a^18-28628104018984*a^17
+62261757956409*a^16+2067346063311366*a^15
-33646881858167757*a^14+15872125201726692*a^13
+31165731682124373*a^12+3118730624434774902*a^11
+19424644835781405573*a^10
+34928604068980953168*a^9-38639423702912510133*a^8
+1269187385301553504698*a^7
+3667076703510023078577*a^6
-455062882588288189452*a^5
-6073679019448320564969*a^4
+39452666543681343198264*a^3
+47232807378617176983216*a^2
+172758700340547215707584*a
+55253832604158366243216)
/79875691357815936000000,
-((89591685911602852*a^19+2456426401544275*a^18-79829287283577388*a^17
-111266747126642685*a^16
+6693211742640744216*a^15
-85247275218262553049*a^14
-7299844963479831120*a^13
+74488279542565111191*a^12
+8274877365995836629420*a^11
+53413178567062212453753*a^10
+121592356188979166775396*a^9
-50703322221226476748215*a^8
+3190417858093889386657296*a^7
+11697039806926436578278045*a^6
+5565042124477506830126808*a^5
-21907925941416463634269203*a^4
+111830991479469725159092680*a^3
+226384145027764567538132592*a^2
+505623402016123027927501440*a
+199640907190542149950414896)
/278840229388945773235200000),
(87530181817718523*a^19+1215087725977030782*a^18
-953928919736667912*a^17
-1313560049466028133*a^16
+9462158957065791483*a^15
-30528725605743381606*a^14
-1150352116504017502104*a^13
+734949752502164464839*a^12
+9542188140971754165201*a^11
+159271468071468573427074*a^10
+789030615124104571528344*a^9
+1002612772932828989646321*a^8
+1441683814063122556892289*a^7
+53856342854315366109409926*a^6
+116471037468993801002554824*a^5
-71683288419330233672409027*a^4
-112235865378225651840900168*a^3
+1132885879666976209353768288*a^2
+1236791680392483380748206592*a
+5963360303581465557565858608)
/1917026577049002190992000000,
-((87530181817718523*a^19+1215087725977030782*a^18
-953928919736667912*a^17
-1313560049466028133*a^16
+9462158957065791483*a^15
-30528725605743381606*a^14
-1150352116504017502104*a^13
+734949752502164464839*a^12
+9542188140971754165201*a^11
+159271468071468573427074*a^10
+789030615124104571528344*a^9
+1002612772932828989646321*a^8
+1441683814063122556892289*a^7
+53856342854315366109409926*a^6
+116471037468993801002554824*a^5
-71683288419330233672409027*a^4
-112235865378225651840900168*a^3
+1132885879666976209353768288*a^2
-680234896656518810243793408*a
+5963360303581465557565858608)
/3834053154098004381984000000)]$
/* Strangely this slightly different example of Dummit has a Galois group of order 10
and is obtained much faster. */
block([s,v],s:splitfield(x^5-5*x+12,x),v:listofvars(s)[1],subst(a,v,s));
[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
+196036,
-((4402021*a^9+8967410*a^8-71413351*a^7-408236968*a^6+199818827*a^5
+3092804186*a^4+4100015915*a^3-6578347876*a^2
+173297643052*a-409949875488)
/34533412800),
(129271*a^9-1030711*a^8-6736419*a^7+9580539*a^6+215434401*a^5
+342066891*a^4-1374973581*a^3-3638935299*a^2+647047544*a
-49872596204)
/34533412800,
(10382*a^9+27879*a^8-140650*a^7-1021019*a^6-626422*a^5+5112817*a^4
+14638214*a^3+12817495*a^2+373402956*a-783084692)
/178929600,
(1134512*a^9+2308737*a^8-18765741*a^7-110380420*a^6+52641936*a^5
+881981807*a^4+1324907097*a^3-2706594556*a^2+41658559300*a
-104470966864)
/8633353200,
-((1134512*a^9+2308737*a^8-18765741*a^7-110380420*a^6+52641936*a^5
+881981807*a^4+1324907097*a^3-2706594556*a^2
+33025206100*a-104470966864)
/17266706400)]$
/* And this one is even more degenerate */
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));
[a^5-110*a^3-55*a^2+2310*a+979,(a^3-3*a^2-72*a+99)/25,
(a^4-4*a^3-94*a^2+196*a+1276)/125,-((a^4+a^3-84*a^2-89*a+671)/125),
(a^2-2*a-44)/5,a];
/* Since splitfield extensively uses algnorm and factorization in extensions, one can
assume that they work correctly */
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