def test_cyclotomic_poly():
pytest.raises(ValueError, lambda: cyclotomic_poly(0, x))
assert cyclotomic_poly(1, x, polys=True) == Poly(x - 1)
assert cyclotomic_poly(1, polys=True) == Poly(x - 1)
assert cyclotomic_poly(1, x) == x - 1
assert cyclotomic_poly(2, x) == x + 1
assert cyclotomic_poly(3, x) == x**2 + x + 1
assert cyclotomic_poly(4, x) == x**2 + 1
assert cyclotomic_poly(5, x) == x**4 + x**3 + x**2 + x + 1
assert cyclotomic_poly(6, x) == x**2 - x + 1
def test_cyclotomic_poly():
pytest.raises(ValueError, lambda: cyclotomic_poly(0, x))
assert cyclotomic_poly(1, x, polys=True) == Poly(x - 1)
assert cyclotomic_poly(1, polys=True) == Poly(x - 1)
assert cyclotomic_poly(1, x) == x - 1
assert cyclotomic_poly(2, x) == x + 1
assert cyclotomic_poly(3, x) == x**2 + x + 1
assert cyclotomic_poly(4, x) == x**2 + 1
assert cyclotomic_poly(5, x) == x**4 + x**3 + x**2 + x + 1
assert cyclotomic_poly(6, x) == x**2 - x + 1
def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in range(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f == g:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
# get the indices in the right order so the computed
# roots will be sorted
h = n//2
ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
d = 2*I*pi/n
for k in reversed(ks):
roots.append(exp(k*d).expand(complex=True))
else:
g = Poly(f, extension=root(-1, n))
for h, _ in ordered(g.factor_list()[1]):
roots.append(-h.TC())
return roots
def _minpoly_exp(ex, x):
"""
Returns the minimal polynomial of ``exp(ex)``
"""
c, a = ex.exp.as_coeff_Mul()
p = sympify(c.p)
q = sympify(c.q)
if a == I * pi:
if c.is_rational:
if c.p == 1 or c.p == -1:
if q == 3:
return x**2 - x + 1
if q == 4:
return x**4 + 1
if q == 6:
return x**4 - x**2 + 1
if q == 8:
return x**8 + 1
if q == 9:
return x**6 - x**3 + 1
if q == 10:
return x**8 - x**6 + x**4 - x**2 + 1
if q.is_prime:
s = 0
for i in range(q):
s += (-x)**i
return s
# x**(2*q) = product(factors)
factors = [cyclotomic_poly(i, x) for i in divisors(2 * q)]
mp = _choose_factor(factors, x, ex)
return mp
else:
raise NotAlgebraic("%s doesn't seem to be an algebraic element" %
ex)
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)