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authorDavid A. Madore <david@procyon>2012-03-15 15:13:27 (GMT)
committerDavid A. Madore <david@procyon>2012-03-15 15:13:27 (GMT)
commit94e0454a6d6d7491286b2c19ed487936835fc7ec (patch)
treeeb553521987d01b7c886ad59114372c3274eee95
parent2dec55d27d7a2bc4b141d49cdb26c6da4ca8478a (diff)
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[radicaux] Expression brute de cos(2π/19).
-rw-r--r--chapitres/radicaux.tex19
-rw-r--r--divers/sageries/racine-19e-de-118
2 files changed, 37 insertions, 0 deletions
diff --git a/chapitres/radicaux.tex b/chapitres/radicaux.tex
index 3ba4931..0e5a9c6 100644
--- a/chapitres/radicaux.tex
+++ b/chapitres/radicaux.tex
@@ -725,6 +725,25 @@ vaut $\sqrt{-11}$. On obtient finalement :
\end{array}
\]
+\subsubsection{$n=19$} \XXX
+
+\[
+\tiny
+\begin{array}{rl}
+\displaystyle\cos\frac{2\pi}{19}
+&\displaystyle=
+-\frac{1}{18}
++(-\frac{1}{36}+\frac{1}{36}\,\sqrt{-3})\,\root 3\of{\frac{133}{2}-\frac{57}{2}\,\sqrt{-3}}
++(-\frac{1}{36}-\frac{1}{36}\,\sqrt{-3})\,\root 3\of{\frac{133}{2}+\frac{57}{2}\,\sqrt{-3}}\\
+&\displaystyle +(-\frac{1}{18}\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}+\frac{1}{18}\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}})\,\root 9\of{\frac{130435}{2}+\frac{144609}{2}\,\sqrt{-3}-214776\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}-1019502\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}+599184\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}+156978\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}\\
+&\displaystyle +(-\frac{1}{18}\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}+\frac{1}{18}\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}})\,\root 9\of{\frac{130435}{2}-\frac{144609}{2}\,\sqrt{-3}+778050\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}+256158\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}-99180\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}-178866\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}\\
+&\displaystyle +\frac{1}{18}\,\root 9\of{\frac{130435}{2}+\frac{144609}{2}\,\sqrt{-3}-41382\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}+241452\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}-420318\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}-57798\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}\\
+&\displaystyle +\frac{1}{18}\,\root 9\of{\frac{130435}{2}-\frac{144609}{2}\,\sqrt{-3}+241452\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}-41382\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}-57798\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}-420318\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}\\
+&\displaystyle +(-\frac{1}{18}\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}+\frac{1}{18}\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}})\,\root 9\of{\frac{130435}{2}+\frac{144609}{2}\,\sqrt{-3}+256158\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}+778050\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}-178866\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}-99180\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}\\
+&\displaystyle +(-\frac{1}{18}\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}+\frac{1}{18}\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}})\,\root 9\of{\frac{130435}{2}-\frac{144609}{2}\,\sqrt{-3}-1019502\,\root 3\of{\frac{1}{2}+\frac{1}{2}\,\sqrt{-3}}-214776\,\root 3\of{\frac{1}{2}-\frac{1}{2}\,\sqrt{-3}}+156978\,\root 3\of{\frac{9}{2}+\frac{3}{2}\,\sqrt{-3}}+599184\,\root 3\of{\frac{9}{2}-\frac{3}{2}\,\sqrt{-3}}}
+\end{array}
+\]
+
\ifx\danslelivre\undefined
diff --git a/divers/sageries/racine-19e-de-1 b/divers/sageries/racine-19e-de-1
new file mode 100644
index 0000000..b9aa12c
--- /dev/null
+++ b/divers/sageries/racine-19e-de-1
@@ -0,0 +1,18 @@
+K.<a> = CyclotomicField(342)
+omega = a^18
+zeta = a^19
+alpha = [sum([zeta^(i*j)*omega^(2^i) for i in range(18)]) for j in range(18)]
+powtab = [NN(18/gcd(i,18)) for i in range(18)]
+atab = [alpha[i]^powtab[i] for i in range(18)]
+atab_on_zeta_basis = [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in atab]
+sqrtm3 = 2*zeta^3-1
+nice_basis = [1, sqrtm3, zeta, zeta^-1, -sqrtm3*zeta^5, sqrtm3*zeta^-5]
+m = Matrix(QQ, 6, 6, [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in nice_basis])
+atab_on_nice_basis = [v * m.inverse() for v in atab_on_zeta_basis]
+zetab = [ZZ(floor(arg(CC(N(alpha[i])/N(atab[i]^(1/powtab[i]))))/arg(zeta)+0.5)) for i in range(18)]
+btab = [zeta^zetab[i] for i in range(18)]
+btab_on_zeta_basis = [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in btab]
+btab_on_nice_basis = [v * m.inverse() for v in btab_on_zeta_basis]
+symbolic_basis = [1, sqrt(-3), ((1/2)*(1+sqrt(-3)))^(1/3), ((1/2)*(1-sqrt(-3)))^(1/3), ((1/2)*(9+3*sqrt(-3)))^(1/3), ((1/2)*(9-3*sqrt(-3)))^(1/3)]
+symbolic_omega = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)]))^(1/powtab[i]) for i in range(18)])/18
+symbolic_cos = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)]))^(1/powtab[i]) for i in range(0,18,2)])/18