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| -rw-r--r-- | chapitres/radicaux.tex | 11 | ||||
| -rw-r--r-- | divers/sageries/racine-13e-de-1 | 18 |
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 @@ vaut $\sqrt{-11}$. On obtient finalement : \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 |