def test_decomp_7():
# Try working through an AlgebraicField
T = Poly(x**3 + x**2 - 2 * x + 8)
K = QQ.algebraic_field((T, theta))
p = 2
P = K.primes_above(p)
ZK = K.maximal_order()
assert len(P) == 3
assert all(Pi.e == Pi.f == 1 for Pi in P)
assert prod(Pi**Pi.e for Pi in P) == p * ZK
def test_solve_triangualted():
f_1 = x**2 + y + z - 1
f_2 = x + y**2 + z - 1
f_3 = x + y + z**2 - 1
a, b = -sqrt(2) - 1, sqrt(2) - 1
assert solve_triangulated([f_1, f_2, f_3], x, y, z) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0)]
dom = QQ.algebraic_field(sqrt(2))
assert solve_triangulated([f_1, f_2, f_3], x, y, z, domain=dom) == \
[(a, a, a), (0, 0, 1), (0, 1, 0), (b, b, b), (1, 0, 0)]
def test_solve_triangualted():
f_1 = x**2 + y + z - 1
f_2 = x + y**2 + z - 1
f_3 = x + y + z**2 - 1
a, b = -sqrt(2) - 1, sqrt(2) - 1
assert solve_triangulated([f_1, f_2, f_3], x, y, z) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0)]
dom = QQ.algebraic_field(sqrt(2))
assert solve_triangulated([f_1, f_2, f_3], x, y, z, domain=dom) == \
[(a, a, a), (0, 0, 1), (0, 1, 0), (b, b, b), (1, 0, 0)]
def test_PrimeIdeal_reduce_poly():
T = Poly(cyclotomic_poly(7, x))
k = QQ.algebraic_field((T, x))
P = k.primes_above(11)
frp = P[0]
B = k.integral_basis(fmt='sympy')
assert [frp._reduce_poly(b, x) for b in B] == [
1, x, x**2, -5 * x**2 - 4 * x + 1, -x**2 - x - 5, 4 * x**2 - x - 1
]
Q = k.primes_above(19)
frq = Q[0]
assert frq.alpha.equiv(0)
assert frq._reduce_poly(20 * x**2 + 10) == x**2 - 9
raises(GeneratorsNeeded, lambda: frp._reduce_poly(S(1)))
raises(NotImplementedError, lambda: frp._reduce_poly(1))
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.algebraic_field((f, theta))
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
