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authorDavid A. Madore <david@procyon.(none)>2011-06-22 18:23:12 +0200
committerDavid A. Madore <david@procyon.(none)>2011-06-22 18:23:12 +0200
commitbefb691798a765506066c0f589056d6b0b5b5a31 (patch)
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parent60b9d8e60bbc32319871bcf41eb4629cc6396d46 (diff)
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[calculs] Résolvante sextique d'un polynôme de degré 5 (pour tester si le groupe de Galois est inclus dans M_20).
Bon, il va falloir arrêter le concours de gros polynômes explicites, à la fin, quand même.
-rw-r--r--chapitres/calculs-galois.tex112
-rw-r--r--divers/sageries/resolvante-m2013
2 files changed, 125 insertions, 0 deletions
diff --git a/chapitres/calculs-galois.tex b/chapitres/calculs-galois.tex
index bd1c70c..6cf0470 100644
--- a/chapitres/calculs-galois.tex
+++ b/chapitres/calculs-galois.tex
@@ -2351,6 +2351,118 @@ existe cependant deux extensions non-isomorphes de $C_4$ par $C_5$ :
celle dont nous parlons est la seule qui soit incluse
dans $\mathfrak{S}_5$.)
+Un polynôme dont le stabilisateur dans $\mathfrak{S}_5$ est $M_{20}$
+est donné par : $P = Z_1^2(Z_2 Z_5 + Z_3 Z_4) + Z_2^2(Z_1 Z_3 + Z_4
+Z_5) + Z_3^2(Z_1 Z_5 + Z_2 Z_4) + Z_4^2(Z_1 Z_2 + Z_3 Z_5) + Z_5^2(Z_1
+Z_4 + Z_2 Z_3)$. La résolvante générale correspondante est : $R_P =
+X^6 + (- 2 a_1 a_3 + 8 a_4) X^5 + (- 8 a_1^3 a_5 + 2 a_1^2 a_2 a_4 +
+a_1^2 a_3^2 + 30 a_1 a_2 a_5 - 14 a_1 a_3 a_4 - 6 a_2^2 a_4 + 2 a_2
+a_3^2 - 50 a_3 a_5 + 40 a_4^2) X^4 + (16 a_1^4 a_3 a_5 - 2 a_1^4 a_4^2
+- 2 a_1^3 a_2^2 a_5 - 2 a_1^3 a_2 a_3 a_4 - 44 a_1^3 a_4 a_5 - 66
+a_1^2 a_2 a_3 a_5 + 21 a_1^2 a_2 a_4^2 + 6 a_1^2 a_3^2 a_4 + 9 a_1
+a_2^3 a_5 + 5 a_1 a_2^2 a_3 a_4 - 2 a_1 a_2 a_3^3 - 50 a_1^2 a_5^2 +
+190 a_1 a_2 a_4 a_5 + 120 a_1 a_3^2 a_5 - 80 a_1 a_3 a_4^2 - 15 a_2^2
+a_3 a_5 - 40 a_2^2 a_4^2 + 21 a_2 a_3^2 a_4 - 2 a_3^4 + 125 a_2 a_5^2
+- 400 a_3 a_4 a_5 + 160 a_4^3) X^3 + (16 a_1^6 a_5^2 - 8 a_1^5 a_2 a_4
+a_5 - 8 a_1^5 a_3^2 a_5 + 2 a_1^5 a_3 a_4^2 + 2 a_1^4 a_2^2 a_3 a_5 +
+a_1^4 a_2^2 a_4^2 - 120 a_1^4 a_2 a_5^2 + 68 a_1^4 a_3 a_4 a_5 - 8
+a_1^4 a_4^3 + 46 a_1^3 a_2^2 a_4 a_5 + 28 a_1^3 a_2 a_3^2 a_5 - 19
+a_1^3 a_2 a_3 a_4^2 - 9 a_1^2 a_2^3 a_3 a_5 - 6 a_1^2 a_2^3 a_4^2 + 3
+a_1^2 a_2^2 a_3^2 a_4 + 250 a_1^3 a_3 a_5^2 - 144 a_1^3 a_4^2 a_5 +
+225 a_1^2 a_2^2 a_5^2 - 354 a_1^2 a_2 a_3 a_4 a_5 + 76 a_1^2 a_2 a_4^3
+- 70 a_1^2 a_3^3 a_5 + 41 a_1^2 a_3^2 a_4^2 - 54 a_1 a_2^3 a_4 a_5 +
+45 a_1 a_2^2 a_3^2 a_5 + 30 a_1 a_2^2 a_3 a_4^2 - 19 a_1 a_2 a_3^3 a_4
++ 2 a_1 a_3^5 + 9 a_2^4 a_4^2 - 6 a_2^3 a_3^2 a_4 + a_2^2 a_3^4 - 200
+a_1^2 a_4 a_5^2 - 875 a_1 a_2 a_3 a_5^2 + 640 a_1 a_2 a_4^2 a_5 + 630
+a_1 a_3^2 a_4 a_5 - 264 a_1 a_3 a_4^3 + 90 a_2^2 a_3 a_4 a_5 - 136
+a_2^2 a_4^3 - 50 a_2 a_3^3 a_5 + 76 a_2 a_3^2 a_4^2 - 8 a_3^4 a_4 +
+500 a_2 a_4 a_5^2 + 625 a_3^2 a_5^2 - 1400 a_3 a_4^2 a_5 + 400 a_4^4)
+X^2 + (- 32 a_1^7 a_3 a_5^2 + 8 a_1^7 a_4^2 a_5 + 8 a_1^6 a_2^2 a_5^2
++ 8 a_1^6 a_2 a_3 a_4 a_5 - 2 a_1^6 a_2 a_4^3 - 2 a_1^5 a_2^3 a_4 a_5
++ 48 a_1^6 a_4 a_5^2 + 264 a_1^5 a_2 a_3 a_5^2 - 94 a_1^5 a_2 a_4^2
+a_5 - 24 a_1^5 a_3^2 a_4 a_5 + 6 a_1^5 a_3 a_4^3 - 66 a_1^4 a_2^3
+a_5^2 - 50 a_1^4 a_2^2 a_3 a_4 a_5 + 19 a_1^4 a_2^2 a_4^3 + 8 a_1^4
+a_2 a_3^3 a_5 - 2 a_1^4 a_2 a_3^2 a_4^2 + 15 a_1^3 a_2^4 a_4 a_5 - 2
+a_1^3 a_2^3 a_3^2 a_5 - a_1^3 a_2^3 a_3 a_4^2 - 56 a_1^5 a_5^3 - 318
+a_1^4 a_2 a_4 a_5^2 - 352 a_1^4 a_3^2 a_5^2 + 166 a_1^4 a_3 a_4^2 a_5
++ 3 a_1^4 a_4^4 - 574 a_1^3 a_2^2 a_3 a_5^2 + 347 a_1^3 a_2^2 a_4^2
+a_5 + 194 a_1^3 a_2 a_3^2 a_4 a_5 - 89 a_1^3 a_2 a_3 a_4^3 - 8 a_1^3
+a_3^4 a_5 + 4 a_1^3 a_3^3 a_4^2 + 162 a_1^2 a_2^4 a_5^2 + 33 a_1^2
+a_2^3 a_3 a_4 a_5 - 51 a_1^2 a_2^3 a_4^3 - 32 a_1^2 a_2^2 a_3^3 a_5 +
+28 a_1^2 a_2^2 a_3^2 a_4^2 - 2 a_1^2 a_2 a_3^4 a_4 - 27 a_1 a_2^5 a_4
+a_5 + 9 a_1 a_2^4 a_3^2 a_5 + 3 a_1 a_2^4 a_3 a_4^2 - a_1 a_2^3 a_3^3
+a_4 + 350 a_1^3 a_2 a_5^3 + 1090 a_1^3 a_3 a_4 a_5^2 - 364 a_1^3 a_4^3
+a_5 + 305 a_1^2 a_2^2 a_4 a_5^2 + 1340 a_1^2 a_2 a_3^2 a_5^2 - 901
+a_1^2 a_2 a_3 a_4^2 a_5 + 76 a_1^2 a_2 a_4^4 - 234 a_1^2 a_3^3 a_4 a_5
++ 102 a_1^2 a_3^2 a_4^3 + 180 a_1 a_2^3 a_3 a_5^2 - 366 a_1 a_2^3
+a_4^2 a_5 - 231 a_1 a_2^2 a_3^2 a_4 a_5 + 212 a_1 a_2^2 a_3 a_4^3 +
+112 a_1 a_2 a_3^4 a_5 - 89 a_1 a_2 a_3^3 a_4^2 + 6 a_1 a_3^5 a_4 - 108
+a_2^5 a_5^2 + 117 a_2^4 a_3 a_4 a_5 + 32 a_2^4 a_4^3 - 31 a_2^3 a_3^3
+a_5 - 51 a_2^3 a_3^2 a_4^2 + 19 a_2^2 a_3^4 a_4 - 2 a_2 a_3^6 - 750
+a_1^2 a_3 a_5^3 - 550 a_1^2 a_4^2 a_5^2 - 375 a_1 a_2^2 a_5^3 - 3075
+a_1 a_2 a_3 a_4 a_5^2 + 1640 a_1 a_2 a_4^3 a_5 - 850 a_1 a_3^3 a_5^2 +
+1220 a_1 a_3^2 a_4^2 a_5 - 384 a_1 a_3 a_4^4 + 525 a_2^3 a_4 a_5^2 -
+325 a_2^2 a_3^2 a_5^2 + 260 a_2^2 a_3 a_4^2 a_5 - 256 a_2^2 a_4^4 +
+105 a_2 a_3^3 a_4 a_5 + 76 a_2 a_3^2 a_4^3 - 58 a_3^5 a_5 + 3 a_3^4
+a_4^2 + 2500 a_1 a_4 a_5^3 + 625 a_2 a_3 a_5^3 - 500 a_2 a_4^2 a_5^2 +
+2750 a_3^2 a_4 a_5^2 - 2400 a_3 a_4^3 a_5 + 512 a_4^5 - 3125 a_5^4) X
++ 16 a_1^8 a_3^2 a_5^2 - 8 a_1^8 a_3 a_4^2 a_5 + a_1^8 a_4^4 - 8 a_1^7
+a_2^2 a_3 a_5^2 + 2 a_1^7 a_2^2 a_4^2 a_5 + a_1^6 a_2^4 a_5^2 - 48
+a_1^7 a_3 a_4 a_5^2 + 12 a_1^7 a_4^3 a_5 + 12 a_1^6 a_2^2 a_4 a_5^2 -
+144 a_1^6 a_2 a_3^2 a_5^2 + 88 a_1^6 a_2 a_3 a_4^2 a_5 - 13 a_1^6 a_2
+a_4^4 + 72 a_1^5 a_2^3 a_3 a_5^2 - 22 a_1^5 a_2^3 a_4^2 a_5 - 4 a_1^5
+a_2^2 a_3^2 a_4 a_5 + a_1^5 a_2^2 a_3 a_4^3 - 9 a_1^4 a_2^5 a_5^2 +
+a_1^4 a_2^4 a_3 a_4 a_5 + 56 a_1^6 a_3 a_5^3 + 86 a_1^6 a_4^2 a_5^2 -
+14 a_1^5 a_2^2 a_5^3 + 304 a_1^5 a_2 a_3 a_4 a_5^2 - 148 a_1^5 a_2
+a_4^3 a_5 + 152 a_1^5 a_3^3 a_5^2 - 54 a_1^5 a_3^2 a_4^2 a_5 + 5 a_1^5
+a_3 a_4^4 - 76 a_1^4 a_2^3 a_4 a_5^2 + 370 a_1^4 a_2^2 a_3^2 a_5^2 -
+287 a_1^4 a_2^2 a_3 a_4^2 a_5 + 65 a_1^4 a_2^2 a_4^4 - 28 a_1^4 a_2
+a_3^3 a_4 a_5 + 5 a_1^4 a_2 a_3^2 a_4^3 + 8 a_1^4 a_3^5 a_5 - 2 a_1^4
+a_3^4 a_4^2 - 210 a_1^3 a_2^4 a_3 a_5^2 + 76 a_1^3 a_2^4 a_4^2 a_5 +
+43 a_1^3 a_2^3 a_3^2 a_4 a_5 - 15 a_1^3 a_2^3 a_3 a_4^3 - 6 a_1^3
+a_2^2 a_3^4 a_5 + 2 a_1^3 a_2^2 a_3^3 a_4^2 + 27 a_1^2 a_2^6 a_5^2 - 9
+a_1^2 a_2^5 a_3 a_4 a_5 + a_1^2 a_2^5 a_4^3 + a_1^2 a_2^4 a_3^3 a_5 -
+468 a_1^5 a_4 a_5^3 - 200 a_1^4 a_2 a_3 a_5^3 - 294 a_1^4 a_2 a_4^2
+a_5^2 - 676 a_1^4 a_3^2 a_4 a_5^2 + 180 a_1^4 a_3 a_4^3 a_5 + 17 a_1^4
+a_4^5 + 50 a_1^3 a_2^3 a_5^3 - 397 a_1^3 a_2^2 a_3 a_4 a_5^2 + 514
+a_1^3 a_2^2 a_4^3 a_5 - 700 a_1^3 a_2 a_3^3 a_5^2 + 447 a_1^3 a_2
+a_3^2 a_4^2 a_5 - 118 a_1^3 a_2 a_3 a_4^4 - 12 a_1^3 a_3^4 a_4 a_5 + 6
+a_1^3 a_3^3 a_4^3 + 141 a_1^2 a_2^4 a_4 a_5^2 - 185 a_1^2 a_2^3 a_3^2
+a_5^2 + 168 a_1^2 a_2^3 a_3 a_4^2 a_5 - 128 a_1^2 a_2^3 a_4^4 + 93
+a_1^2 a_2^2 a_3^3 a_4 a_5 + 19 a_1^2 a_2^2 a_3^2 a_4^3 - 36 a_1^2 a_2
+a_3^5 a_5 + 5 a_1^2 a_2 a_3^4 a_4^2 + 198 a_1 a_2^5 a_3 a_5^2 - 78 a_1
+a_2^5 a_4^2 a_5 - 95 a_1 a_2^4 a_3^2 a_4 a_5 + 44 a_1 a_2^4 a_3 a_4^3
++ 25 a_1 a_2^3 a_3^4 a_5 - 15 a_1 a_2^3 a_3^3 a_4^2 + a_1 a_2^2 a_3^5
+a_4 - 27 a_2^7 a_5^2 + 18 a_2^6 a_3 a_4 a_5 - 4 a_2^6 a_4^3 - 4 a_2^5
+a_3^3 a_5 + a_2^5 a_3^2 a_4^2 + 625 a_1^4 a_5^4 + 2300 a_1^3 a_2 a_4
+a_5^3 + 250 a_1^3 a_3^2 a_5^3 + 1470 a_1^3 a_3 a_4^2 a_5^2 - 276 a_1^3
+a_4^4 a_5 - 125 a_1^2 a_2^2 a_3 a_5^3 - 610 a_1^2 a_2^2 a_4^2 a_5^2 +
+1995 a_1^2 a_2 a_3^2 a_4 a_5^2 - 1174 a_1^2 a_2 a_3 a_4^3 a_5 - 16
+a_1^2 a_2 a_4^5 + 375 a_1^2 a_3^4 a_5^2 - 172 a_1^2 a_3^3 a_4^2 a_5 +
+82 a_1^2 a_3^2 a_4^4 + 15 a_1 a_2^3 a_3 a_4 a_5^2 - 384 a_1 a_2^3
+a_4^3 a_5 + 525 a_1 a_2^2 a_3^3 a_5^2 - 528 a_1 a_2^2 a_3^2 a_4^2 a_5
++ 384 a_1 a_2^2 a_3 a_4^4 - 29 a_1 a_2 a_3^4 a_4 a_5 - 118 a_1 a_2
+a_3^3 a_4^3 + 38 a_1 a_3^6 a_5 + 5 a_1 a_3^5 a_4^2 - 99 a_2^5 a_4
+a_5^2 - 150 a_2^4 a_3^2 a_5^2 + 196 a_2^4 a_3 a_4^2 a_5 + 48 a_2^4
+a_4^4 + 12 a_2^3 a_3^3 a_4 a_5 - 128 a_2^3 a_3^2 a_4^3 - 12 a_2^2
+a_3^5 a_5 + 65 a_2^2 a_3^4 a_4^2 - 13 a_2 a_3^6 a_4 + a_3^8 - 3125
+a_1^2 a_2 a_5^4 - 3500 a_1^2 a_3 a_4 a_5^3 - 1450 a_1^2 a_4^3 a_5^2 -
+1750 a_1 a_2^2 a_4 a_5^3 + 625 a_1 a_2 a_3^2 a_5^3 - 850 a_1 a_2 a_3
+a_4^2 a_5^2 + 1760 a_1 a_2 a_4^4 a_5 - 2050 a_1 a_3^3 a_4 a_5^2 + 780
+a_1 a_3^2 a_4^3 a_5 - 192 a_1 a_3 a_4^5 + 1200 a_2^3 a_4^2 a_5^2 - 725
+a_2^2 a_3^2 a_4 a_5^2 - 160 a_2^2 a_3 a_4^3 a_5 - 192 a_2^2 a_4^5 -
+125 a_2 a_3^4 a_5^2 + 590 a_2 a_3^3 a_4^2 a_5 - 16 a_2 a_3^2 a_4^4 -
+124 a_3^5 a_4 a_5 + 17 a_3^4 a_4^3 + 3125 a_1 a_3 a_5^4 + 7500 a_1
+a_4^2 a_5^3 + 3125 a_2^2 a_5^4 - 1250 a_2 a_3 a_4 a_5^3 - 2000 a_2
+a_4^3 a_5^2 + 3250 a_3^2 a_4^2 a_5^2 - 1600 a_3 a_4^4 a_5 + 256 a_4^6
+- 9375 a_4 a_5^4$. Cette résolvante sextique admet donc une racine
+si, et lorsqu'elle est séparable seulement si, le polynôme $f = X^5 +
+a_1 X^4 + a_2 X^3 + a_3 X^2 + a_4 X + a_5$ (supposé irréductible et
+séparable) a un groupe de Galois inclus dans $M_{20}$.
+
+\XXX En fait, $R_P$ ne peut pas être réductible autrement qu'en ayant
+une racine.
+
\ifx\danslelivre\undefined
diff --git a/divers/sageries/resolvante-m20 b/divers/sageries/resolvante-m20
new file mode 100644
index 0000000..15e18c0
--- /dev/null
+++ b/divers/sageries/resolvante-m20
@@ -0,0 +1,13 @@
+e = SymmetricFunctionAlgebra(QQ, basis='elementary')
+R.<x,z1,z2,z3,z4,z5,a1,a2,a3,a4,a5> = PolynomialRing(QQ,11,order='lex(1),lex(5),deglex(5)')
+asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
+asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
+asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
+asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
+asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
+Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5])
+p = z1^2*(z2*z5 + z3*z4) + z2^2*(z1*z3 + z4*z5) + z3^2*(z1*z5 + z2*z4) + z4^2*(z1*z2 + z3*z5) + z5^2*(z1*z4 + z2*z3)
+tmp = (x-p)*(x-p).subs({z4:z5,z5:z4})
+res = tmp*tmp.subs({z1:z2,z2:z3,z3:z1})*tmp.subs({z1:z3,z2:z1,z3:z2})
+B = Isym.groebner_basis()
+res0 = res.reduce(B)