def _construct_algebraic(coeffs, opt):
"""We know that coefficients are algebraic so construct the extension. """
from sympy.polys.numberfields import primitive_element
exts = set()
def build_trees(args):
trees = []
for a in args:
if a.is_Rational:
tree = ('Q', QQ.from_sympy(a))
elif a.is_Add:
tree = ('+', build_trees(a.args))
elif a.is_Mul:
tree = ('*', build_trees(a.args))
else:
tree = ('e', a)
exts.add(a)
trees.append(tree)
return trees
trees = build_trees(coeffs)
exts = list(ordered(exts))
g, span, H = primitive_element(exts, ex=True, polys=True)
root = sum([s * ext for s, ext in zip(span, exts)])
domain, g = QQ.algebraic_field((g, root)), g.rep.rep
exts_dom = [domain.dtype.from_list(h, g, QQ) for h in H]
exts_map = dict(zip(exts, exts_dom))
def convert_tree(tree):
op, args = tree
if op == 'Q':
return domain.dtype.from_list([args], g, QQ)
elif op == '+':
return sum((convert_tree(a) for a in args), domain.zero)
elif op == '*':
# return prod(convert(a) for a in args)
t = convert_tree(args[0])
for a in args[1:]:
t *= convert_tree(a)
return t
elif op == 'e':
return exts_map[args]
else:
raise RuntimeError
result = [convert_tree(tree) for tree in trees]
return domain, result
def test_primitive_element():
assert primitive_element([sqrt(2)], x) == (x**2 - 2, [1])
assert primitive_element([sqrt(2), sqrt(3)], x) == (x**4 - 10*x**2 + 1, [1, 1])
assert primitive_element([sqrt(2)], x, polys=True) == (Poly(x**2 - 2), [1])
assert primitive_element([sqrt(2), sqrt(3)], x, polys=True) == (Poly(x**4 - 10*x**2 + 1), [1, 1])
assert primitive_element([sqrt(2)], x, ex=True) == (x**2 - 2, [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True) == \
(x**4 - 10*x**2 + 1, [1, 1], [[Q(1,2), 0, -Q(9,2), 0], [-Q(1,2), 0, Q(11,2), 0]])
assert primitive_element([sqrt(2)], x, ex=True, polys=True) == (Poly(x**2 - 2), [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True, polys=True) == \
(Poly(x**4 - 10*x**2 + 1), [1, 1], [[Q(1,2), 0, -Q(9,2), 0], [-Q(1,2), 0, Q(11,2), 0]])
assert primitive_element([sqrt(2)], polys=True) == (Poly(x**2 - 2), [1])
raises(ValueError, "primitive_element([], x, ex=False)")
raises(ValueError, "primitive_element([], x, ex=True)")
def _construct_algebraic(coeffs, opt):
"""We know that coefficients are algebraic so construct the extension. """
from sympy.polys.numberfields import primitive_element
result, exts = [], set([])
for coeff in coeffs:
if coeff.is_Rational:
coeff = (None, 0, QQ.from_sympy(coeff))
else:
a = coeff.as_coeff_add()[0]
coeff -= a
b = coeff.as_coeff_mul()[0]
coeff /= b
exts.add(coeff)
a = QQ.from_sympy(a)
b = QQ.from_sympy(b)
coeff = (coeff, b, a)
result.append(coeff)
exts = list(exts)
g, span, H = primitive_element(exts, ex=True, polys=True)
root = sum([ s*ext for s, ext in zip(span, exts) ])
domain, g = QQ.algebraic_field((g, root)), g.rep.rep
for i, (coeff, a, b) in enumerate(result):
if coeff is not None:
coeff = a*domain.dtype.from_list(H[exts.index(coeff)], g, QQ) + b
else:
coeff = domain.dtype.from_list([b], g, QQ)
result[i] = coeff
return domain, result
def _construct_algebraic(coeffs, opt):
"""We know that coefficients are algebraic so construct the extension. """
from sympy.polys.numberfields import primitive_element
result, exts = [], set([])
for coeff in coeffs:
if coeff.is_Rational:
coeff = (None, 0, QQ.from_sympy(coeff))
else:
a = coeff.as_coeff_add()[0]
coeff -= a
b = coeff.as_coeff_mul()[0]
coeff /= b
exts.add(coeff)
a = QQ.from_sympy(a)
b = QQ.from_sympy(b)
coeff = (coeff, b, a)
result.append(coeff)
exts = list(exts)
g, span, H = primitive_element(exts, ex=True, polys=True)
root = sum([s * ext for s, ext in zip(span, exts)])
domain, g = QQ.algebraic_field((g, root)), g.rep.rep
for i, (coeff, a, b) in enumerate(result):
if coeff is not None:
coeff = a * domain.dtype.from_list(H[exts.index(coeff)], g, QQ) + b
else:
coeff = domain.dtype.from_list([b], g, QQ)
result[i] = coeff
return domain, result
def test_primitive_element():
assert primitive_element([sqrt(2)], x) == (x**2 - 2, [1])
assert primitive_element([sqrt(2), sqrt(3)], x) == (
x**4 - 10 * x**2 + 1,
[1, 1],
)
assert primitive_element([sqrt(2)], x, polys=True) == (Poly(x**2 - 2), [1])
assert primitive_element([sqrt(2), sqrt(3)], x, polys=True) == (
Poly(x**4 - 10 * x**2 + 1),
[1, 1],
)
assert primitive_element([sqrt(2)], x, ex=True) == (x**2 - 2, [1], [[1,
0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True) == (
x**4 - 10 * x**2 + 1,
[1, 1],
[[Q(1, 2), 0, -Q(9, 2), 0], [-Q(1, 2), 0, Q(11, 2), 0]],
)
assert primitive_element([sqrt(2)], x, ex=True, polys=True) == (
Poly(x**2 - 2),
[1],
[[1, 0]],
)
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True, polys=True) == (
Poly(x**4 - 10 * x**2 + 1),
[1, 1],
[[Q(1, 2), 0, -Q(9, 2), 0], [-Q(1, 2), 0, Q(11, 2), 0]],
)
assert primitive_element([sqrt(2)], polys=True) == (Poly(x**2 - 2), [1])
raises(ValueError, lambda: primitive_element([], x, ex=False))
raises(ValueError, lambda: primitive_element([], x, ex=True))
# Issue 14117
a, b = I * sqrt(2 * sqrt(2) + 3), I * sqrt(-2 * sqrt(2) + 3)
assert primitive_element([a, b, I], x) == (x**4 + 6 * x**2 + 1, [1, 0, 0])
def test_primitive_element():
assert primitive_element([sqrt(2)], x) == (x**2 - 2, [1])
assert primitive_element(
[sqrt(2), sqrt(3)], x) == (x**4 - 10*x**2 + 1, [1, 1])
assert primitive_element([sqrt(2)], x, polys=True) == (Poly(x**2 - 2), [1])
assert primitive_element([sqrt(
2), sqrt(3)], x, polys=True) == (Poly(x**4 - 10*x**2 + 1), [1, 1])
assert primitive_element(
[sqrt(2)], x, ex=True) == (x**2 - 2, [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True) == \
(x**4 - 10*x**2 + 1, [1, 1], [[Q(1, 2), 0, -Q(9, 2), 0], [-
Q(1, 2), 0, Q(11, 2), 0]])
assert primitive_element(
[sqrt(2)], x, ex=True, polys=True) == (Poly(x**2 - 2), [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True, polys=True) == \
(Poly(x**4 - 10*x**2 + 1), [1, 1], [[Q(1, 2), 0, -Q(9, 2),
0], [-Q(1, 2), 0, Q(11, 2), 0]])
assert primitive_element([sqrt(2)], polys=True) == (Poly(x**2 - 2), [1])
raises(ValueError, lambda: primitive_element([], x, ex=False))
raises(ValueError, lambda: primitive_element([], x, ex=True))
# Issue 14117
a, b = I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3)
assert primitive_element([a, b, I], x) == (x**4 + 6*x**2 + 1, [1, 0, 0])