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-rw-r--r--chapitres/radicaux.tex11
-rw-r--r--divers/sageries/racine-13e-de-118
2 files changed, 29 insertions, 0 deletions
diff --git a/chapitres/radicaux.tex b/chapitres/radicaux.tex
index 89be819..5846065 100644
--- a/chapitres/radicaux.tex
+++ b/chapitres/radicaux.tex
@@ -704,6 +704,17 @@ vautĀ $\sqrt{-11}$. On obtient finalementĀ :
\end{array}
\]
+\subsubsection{$n=13$} \XXX
+
+\[
+\begin{array}{rl}
+\displaystyle\cos\frac{2\pi}{13}
+&\displaystyle= - \frac{1}{12} + \frac{1}{12} \, \sqrt{13} + \frac{1}{24} {\left(-1 + \sqrt{-3}\right)} \, \root3\of{-\frac{65}{2} - \frac{39}{2} \, \sqrt{-3}}
+- \frac{1}{24} \, {\left(1 + \sqrt{-3}\right)} \root3\of{-\frac{65}{2} + \frac{39}{2} \, \sqrt{-3}}\\
+&\displaystyle + \frac{1}{24} {\left(1 + \sqrt{-3}\right)} \, \root6\of{-\frac{4381}{2} - \frac{195}{2} \, \sqrt{-3}} - \frac{1}{24} \, {\left(-1 + \sqrt{-3}\right)} \root6\of{-\frac{4381}{2} + \frac{195}{2} \, \sqrt{-3}}\\
+\end{array}
+\]
+
\ifx\danslelivre\undefined
diff --git a/divers/sageries/racine-13e-de-1 b/divers/sageries/racine-13e-de-1
new file mode 100644
index 0000000..e0b40eb
--- /dev/null
+++ b/divers/sageries/racine-13e-de-1
@@ -0,0 +1,18 @@
+K.<a> = CyclotomicField(156)
+omega = a^12
+zeta = a^13
+alpha = [sum([zeta^(i*j)*omega^(2^i) for i in range(12)]) for j in range(13)]
+powtab = [NN(12/gcd(i,12)) for i in range(12)]
+atab = [alpha[i]^powtab[i] for i in range(12)]
+atab_on_zeta_basis = [(QQ^4)((zeta.coordinates_in_terms_of_powers())(x)) for x in atab]
+sqrtm3 = 2*zeta^2-1
+nice_basis = [1, sqrtm3, zeta, -zeta*sqrtm3]
+m = Matrix(QQ, 4, 4, [(QQ^4)((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(12)]
+btab = [zeta^zetab[i] for i in range(12)]
+btab_on_zeta_basis = [(QQ^4)((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), sqrt((1/2)*(1+sqrt(-3))), sqrt(-(3/2)*(1+sqrt(-3)))]
+symbolic_zeta = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)]))^(1/powtab[i]) for i in range(12)])/12
+symbolic_cos = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)]))^(1/powtab[i]) for i in range(0,12,2)])/12