def test_round_two():
# Poly must be monic, irreducible, and over ZZ:
raises(ValueError, lambda: round_two(Poly(3 * x ** 2 + 1)))
raises(ValueError, lambda: round_two(Poly(x ** 2 - 1)))
raises(ValueError, lambda: round_two(Poly(x ** 2 + QQ(1, 2))))
# Test on many fields:
cases = (
# A couple of cyclotomic fields:
(cyclotomic_poly(5), DomainMatrix.eye(4, QQ), 125),
(cyclotomic_poly(7), DomainMatrix.eye(6, QQ), -16807),
# A couple of quadratic fields (one 1 mod 4, one 3 mod 4):
(x ** 2 - 5, DM([[1, (1, 2)], [0, (1, 2)]], QQ), 5),
(x ** 2 - 7, DM([[1, 0], [0, 1]], QQ), 28),
# Dedekind's example of a field with 2 as essential disc divisor:
(x ** 3 + x ** 2 - 2 * x + 8, DM([[1, 0, 0], [0, 1, 0], [0, (1, 2), (1, 2)]], QQ).transpose(), -503),
# A bunch of cubics with various forms for F -- all of these require
# second or third enlargements. (Five of them require a third, while the rest require just a second.)
# F = 2^2
(x**3 + 3 * x**2 - 4 * x + 4, DM([((1, 2), (1, 4), (1, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -83),
# F = 2^2 * 3
(x**3 + 3 * x**2 + 3 * x - 3, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -108),
# F = 2^3
(x**3 + 5 * x**2 - x + 3, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -31),
# F = 2^2 * 5
(x**3 + 5 * x**2 - 5 * x - 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 1300),
# F = 3^2
(x**3 + 3 * x**2 + 5, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -135),
# F = 3^3
(x**3 + 6 * x**2 + 3 * x - 1, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 81),
# F = 2^2 * 3^2
(x**3 + 6 * x**2 + 4, DM([((1, 3), (2, 3), (1, 3)), (0, 1, 0), (0, 0, (1, 2))], QQ).transpose(), -108),
# F = 2^3 * 7
(x**3 + 7 * x**2 + 7 * x - 7, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 49),
# F = 2^2 * 13
(x**3 + 7 * x**2 - x + 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -2028),
# F = 2^4
(x**3 + 7 * x**2 - 5 * x + 5, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -140),
# F = 5^2
(x**3 + 4 * x**2 - 3 * x + 7, DM([((1, 5), (4, 5), (4, 5)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -175),
# F = 7^2
(x**3 + 8 * x**2 + 5 * x - 1, DM([((1, 7), (6, 7), (2, 7)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 49),
# F = 2 * 5 * 7
(x**3 + 8 * x**2 - 2 * x + 6, DM([(1, 0, 0), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -14700),
# F = 2^2 * 3 * 5
(x**3 + 6 * x**2 - 3 * x + 8, DM([(1, 0, 0), (0, (1, 4), (1, 4)), (0, 0, 1)], QQ).transpose(), -675),
# F = 2 * 3^2 * 7
(x**3 + 9 * x**2 + 6 * x - 8, DM([(1, 0, 0), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 3969),
# F = 2^2 * 3^2 * 7
(x**3 + 15 * x**2 - 9 * x + 13, DM([((1, 6), (1, 3), (1, 6)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -5292),
)
for f, B_exp, d_exp in cases:
K = QQ.alg_field_from_poly(f)
B = K.maximal_order().QQ_matrix
d = K.discriminant()
assert d == d_exp
# The computed basis need not equal the expected one, but their quotient
# must be unimodular:
assert (B.inv()*B_exp).det()**2 == 1
def test_Domain_alg_field_from_poly():
f = Poly(x**2 - 2)
g = Poly(x**2 - 3)
h = Poly(x**4 - 10*x**2 + 1)
alg = ZZ.alg_field_from_poly(f)
assert alg.ext.minpoly == f
assert alg.dom == QQ
alg = QQ.alg_field_from_poly(f)
assert alg.ext.minpoly == f
assert alg.dom == QQ
alg = alg.alg_field_from_poly(g)
assert alg.ext.minpoly == h
assert alg.dom == QQ
