[radicaux] Expression brute de cos(2π/17).

1 files changed, 32 insertions, 0 deletions

diff --git a/divers/sageries/racine-17e-de-1 b/divers/sageries/racine-17e-de-1
new file mode 100644
index 0000000..31af5b9
--- /dev/null
+++ b/divers/sageries/racine-17e-de-1
@@ -0,0 +1,32 @@
+K.<omega> = CyclotomicField(17)
+sqrt17 = sum([(-1)^i*omega^(3^i) for i in range(16)])
+sub2 = [sum([(-1)^i*omega^(3^j*9^i) for i in range(8)]) for j in range(2)]
+sub4 = [sum([(-1)^i*omega^(3^j*13^i) for i in range(4)]) for j in range(4)]
+sub8 = [sum([(-1)^i*omega^(3^j*16^i) for i in range(2)]) for j in range(8)]
+nice_basis = [1, sqrt17] + sub2 + sub4 + sub8
+for i in range(16):
+    z = CC(nice_basis[i])
+    if z.real_part()<=-0.0001 or (abs(z.real_part())<0.0001 and z.imag_part()<0):
+        nice_basis[i] = -nice_basis[i]
+
+sub2 = nice_basis[2:4]
+sub4 = nice_basis[4:8]
+sub8 = nice_basis[8:16]
+m = Matrix(QQ, 16, 16, [(QQ^16)((omega.coordinates_in_terms_of_powers())(x)) for x in nice_basis])
+def on_nice_basis(x):
+    return (QQ^16)((omega.coordinates_in_terms_of_powers())(x)) * m.inverse()
+
+symbolic_sqrt17 = sqrt(17)
+symbolic_basis = [1, symbolic_sqrt17]
+def symbolicize(x):
+    v = on_nice_basis(x)
+    return sum([v[i]*symbolic_basis[i] for i in range(16) if v[i]!=0])
+
+symbolic_sub2 = [sqrt(symbolicize(x^2)) for x in sub2]
+symbolic_basis = symbolic_basis + symbolic_sub2
+symbolic_sub4 = [sqrt(symbolicize(x^2)) for x in sub4]
+symbolic_basis = symbolic_basis + symbolic_sub4
+symbolic_sub8 = [sqrt(symbolicize(x^2)) for x in sub8]
+symbolic_basis = symbolic_basis + symbolic_sub8
+symbolic_cos = symbolicize((omega+omega^-1)/2)
+symbolic_omega = symbolicize(omega)