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| 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])] | |