Vehicle-cpp/hkpc/software/sympy/polys/numberfields/tests/test_basis.py

from sympy.abc import x
from sympy.core import S
from sympy.core.numbers import AlgebraicNumber
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.polys import Poly, cyclotomic_poly
from sympy.polys.domains import QQ
from sympy.polys.matrices import DomainMatrix, DM
from sympy.polys.numberfields.basis import round_two
from sympy.testing.pytest import raises


def test_round_two():
    # Poly must be irreducible, and over ZZ or QQ:
    raises(ValueError, lambda: round_two(Poly(x ** 2 - 1)))
    raises(ValueError, lambda: round_two(Poly(x ** 2 + sqrt(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),
        # Polynomial need not be monic
        (5*x**3 + 5*x**2 - 10 * x + 40, DM([[1, 0, 0], [0, 1, 0], [0, (1, 2), (1, 2)]], QQ).transpose(), -503),
        # Polynomial can have non-integer rational coeffs
        (QQ(5, 3)*x**3 + QQ(5, 3)*x**2 - QQ(10, 3)*x + QQ(40, 3), DM([[1, 0, 0], [0, 1, 0], [0, (1, 2), (1, 2)]], QQ).transpose(), -503),
    )
    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_AlgebraicField_integral_basis():
    alpha = AlgebraicNumber(sqrt(5), alias='alpha')
    k = QQ.algebraic_field(alpha)
    B0 = k.integral_basis()
    B1 = k.integral_basis(fmt='sympy')
    B2 = k.integral_basis(fmt='alg')
    assert B0 == [k([1]), k([S.Half, S.Half])]
    assert B1 == [1, S.Half + alpha/2]
    assert B2 == [k.ext.field_element([1]),
                  k.ext.field_element([S.Half, S.Half])]