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298 lines
12 KiB
298 lines
12 KiB
"""Tests for the subfield problem and allied problems. """
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from sympy.core.numbers import (AlgebraicNumber, I, pi, Rational)
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from sympy.core.singleton import S
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.polys.numberfields.subfield import (
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is_isomorphism_possible,
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field_isomorphism_pslq,
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field_isomorphism,
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primitive_element,
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to_number_field,
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)
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from sympy.polys.polyerrors import IsomorphismFailed
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from sympy.polys.polytools import Poly
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from sympy.polys.rootoftools import CRootOf
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from sympy.testing.pytest import raises
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from sympy.abc import x
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Q = Rational
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def test_field_isomorphism_pslq():
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a = AlgebraicNumber(I)
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b = AlgebraicNumber(I*sqrt(3))
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raises(NotImplementedError, lambda: field_isomorphism_pslq(a, b))
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a = AlgebraicNumber(sqrt(2))
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b = AlgebraicNumber(sqrt(3))
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c = AlgebraicNumber(sqrt(7))
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d = AlgebraicNumber(sqrt(2) + sqrt(3))
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e = AlgebraicNumber(sqrt(2) + sqrt(3) + sqrt(7))
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assert field_isomorphism_pslq(a, a) == [1, 0]
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assert field_isomorphism_pslq(a, b) is None
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assert field_isomorphism_pslq(a, c) is None
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assert field_isomorphism_pslq(a, d) == [Q(1, 2), 0, -Q(9, 2), 0]
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assert field_isomorphism_pslq(
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a, e) == [Q(1, 80), 0, -Q(1, 2), 0, Q(59, 20), 0]
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assert field_isomorphism_pslq(b, a) is None
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assert field_isomorphism_pslq(b, b) == [1, 0]
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assert field_isomorphism_pslq(b, c) is None
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assert field_isomorphism_pslq(b, d) == [-Q(1, 2), 0, Q(11, 2), 0]
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assert field_isomorphism_pslq(b, e) == [-Q(
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3, 640), 0, Q(67, 320), 0, -Q(297, 160), 0, Q(313, 80), 0]
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assert field_isomorphism_pslq(c, a) is None
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assert field_isomorphism_pslq(c, b) is None
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assert field_isomorphism_pslq(c, c) == [1, 0]
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assert field_isomorphism_pslq(c, d) is None
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assert field_isomorphism_pslq(c, e) == [Q(
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3, 640), 0, -Q(71, 320), 0, Q(377, 160), 0, -Q(469, 80), 0]
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assert field_isomorphism_pslq(d, a) is None
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assert field_isomorphism_pslq(d, b) is None
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assert field_isomorphism_pslq(d, c) is None
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assert field_isomorphism_pslq(d, d) == [1, 0]
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assert field_isomorphism_pslq(d, e) == [-Q(
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3, 640), 0, Q(71, 320), 0, -Q(377, 160), 0, Q(549, 80), 0]
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assert field_isomorphism_pslq(e, a) is None
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assert field_isomorphism_pslq(e, b) is None
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assert field_isomorphism_pslq(e, c) is None
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assert field_isomorphism_pslq(e, d) is None
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assert field_isomorphism_pslq(e, e) == [1, 0]
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f = AlgebraicNumber(3*sqrt(2) + 8*sqrt(7) - 5)
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assert field_isomorphism_pslq(
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f, e) == [Q(3, 80), 0, -Q(139, 80), 0, Q(347, 20), 0, -Q(761, 20), -5]
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def test_field_isomorphism():
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assert field_isomorphism(3, sqrt(2)) == [3]
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assert field_isomorphism( I*sqrt(3), I*sqrt(3)/2) == [ 2, 0]
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assert field_isomorphism(-I*sqrt(3), I*sqrt(3)/2) == [-2, 0]
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assert field_isomorphism( I*sqrt(3), -I*sqrt(3)/2) == [-2, 0]
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assert field_isomorphism(-I*sqrt(3), -I*sqrt(3)/2) == [ 2, 0]
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assert field_isomorphism( 2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [ Rational(6, 35), 0]
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assert field_isomorphism(-2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [Rational(-6, 35), 0]
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assert field_isomorphism( 2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [Rational(-6, 35), 0]
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assert field_isomorphism(-2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [ Rational(6, 35), 0]
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assert field_isomorphism(
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2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [ Rational(6, 35), 27]
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assert field_isomorphism(
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-2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [Rational(-6, 35), 27]
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assert field_isomorphism(
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2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [Rational(-6, 35), 27]
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assert field_isomorphism(
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-2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [ Rational(6, 35), 27]
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p = AlgebraicNumber( sqrt(2) + sqrt(3))
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q = AlgebraicNumber(-sqrt(2) + sqrt(3))
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r = AlgebraicNumber( sqrt(2) - sqrt(3))
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s = AlgebraicNumber(-sqrt(2) - sqrt(3))
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pos_coeffs = [ S.Half, S.Zero, Rational(-9, 2), S.Zero]
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neg_coeffs = [Rational(-1, 2), S.Zero, Rational(9, 2), S.Zero]
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a = AlgebraicNumber(sqrt(2))
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == pos_coeffs
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assert field_isomorphism(a, q, fast=True) == neg_coeffs
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assert field_isomorphism(a, r, fast=True) == pos_coeffs
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assert field_isomorphism(a, s, fast=True) == neg_coeffs
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assert field_isomorphism(a, p, fast=False) == pos_coeffs
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assert field_isomorphism(a, q, fast=False) == neg_coeffs
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assert field_isomorphism(a, r, fast=False) == pos_coeffs
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assert field_isomorphism(a, s, fast=False) == neg_coeffs
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a = AlgebraicNumber(-sqrt(2))
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == neg_coeffs
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assert field_isomorphism(a, q, fast=True) == pos_coeffs
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assert field_isomorphism(a, r, fast=True) == neg_coeffs
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assert field_isomorphism(a, s, fast=True) == pos_coeffs
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assert field_isomorphism(a, p, fast=False) == neg_coeffs
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assert field_isomorphism(a, q, fast=False) == pos_coeffs
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assert field_isomorphism(a, r, fast=False) == neg_coeffs
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assert field_isomorphism(a, s, fast=False) == pos_coeffs
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pos_coeffs = [ S.Half, S.Zero, Rational(-11, 2), S.Zero]
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neg_coeffs = [Rational(-1, 2), S.Zero, Rational(11, 2), S.Zero]
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a = AlgebraicNumber(sqrt(3))
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == neg_coeffs
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assert field_isomorphism(a, q, fast=True) == neg_coeffs
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assert field_isomorphism(a, r, fast=True) == pos_coeffs
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assert field_isomorphism(a, s, fast=True) == pos_coeffs
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assert field_isomorphism(a, p, fast=False) == neg_coeffs
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assert field_isomorphism(a, q, fast=False) == neg_coeffs
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assert field_isomorphism(a, r, fast=False) == pos_coeffs
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assert field_isomorphism(a, s, fast=False) == pos_coeffs
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a = AlgebraicNumber(-sqrt(3))
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == pos_coeffs
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assert field_isomorphism(a, q, fast=True) == pos_coeffs
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assert field_isomorphism(a, r, fast=True) == neg_coeffs
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assert field_isomorphism(a, s, fast=True) == neg_coeffs
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assert field_isomorphism(a, p, fast=False) == pos_coeffs
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assert field_isomorphism(a, q, fast=False) == pos_coeffs
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assert field_isomorphism(a, r, fast=False) == neg_coeffs
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assert field_isomorphism(a, s, fast=False) == neg_coeffs
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pos_coeffs = [ Rational(3, 2), S.Zero, Rational(-33, 2), -S(8)]
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neg_coeffs = [Rational(-3, 2), S.Zero, Rational(33, 2), -S(8)]
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a = AlgebraicNumber(3*sqrt(3) - 8)
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == neg_coeffs
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assert field_isomorphism(a, q, fast=True) == neg_coeffs
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assert field_isomorphism(a, r, fast=True) == pos_coeffs
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assert field_isomorphism(a, s, fast=True) == pos_coeffs
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assert field_isomorphism(a, p, fast=False) == neg_coeffs
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assert field_isomorphism(a, q, fast=False) == neg_coeffs
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assert field_isomorphism(a, r, fast=False) == pos_coeffs
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assert field_isomorphism(a, s, fast=False) == pos_coeffs
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a = AlgebraicNumber(3*sqrt(2) + 2*sqrt(3) + 1)
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pos_1_coeffs = [ S.Half, S.Zero, Rational(-5, 2), S.One]
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neg_5_coeffs = [Rational(-5, 2), S.Zero, Rational(49, 2), S.One]
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pos_5_coeffs = [ Rational(5, 2), S.Zero, Rational(-49, 2), S.One]
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neg_1_coeffs = [Rational(-1, 2), S.Zero, Rational(5, 2), S.One]
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assert is_isomorphism_possible(a, p) is True
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assert is_isomorphism_possible(a, q) is True
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assert is_isomorphism_possible(a, r) is True
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assert is_isomorphism_possible(a, s) is True
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assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
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assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
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assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
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assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
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assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
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assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
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assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
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assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
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a = AlgebraicNumber(sqrt(2))
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b = AlgebraicNumber(sqrt(3))
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c = AlgebraicNumber(sqrt(7))
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assert is_isomorphism_possible(a, b) is True
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assert is_isomorphism_possible(b, a) is True
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assert is_isomorphism_possible(c, p) is False
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assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
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assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
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assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
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assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
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a = AlgebraicNumber(sqrt(2))
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b = AlgebraicNumber(2 ** (S(1) / 3))
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assert is_isomorphism_possible(a, b) is False
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assert field_isomorphism(a, b) is None
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def test_primitive_element():
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assert primitive_element([sqrt(2)], x) == (x**2 - 2, [1])
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assert primitive_element(
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[sqrt(2), sqrt(3)], x) == (x**4 - 10*x**2 + 1, [1, 1])
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assert primitive_element([sqrt(2)], x, polys=True) == (Poly(x**2 - 2, domain='QQ'), [1])
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assert primitive_element([sqrt(
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2), sqrt(3)], x, polys=True) == (Poly(x**4 - 10*x**2 + 1, domain='QQ'), [1, 1])
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assert primitive_element(
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[sqrt(2)], x, ex=True) == (x**2 - 2, [1], [[1, 0]])
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assert primitive_element([sqrt(2), sqrt(3)], x, ex=True) == \
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(x**4 - 10*x**2 + 1, [1, 1], [[Q(1, 2), 0, -Q(9, 2), 0], [-
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Q(1, 2), 0, Q(11, 2), 0]])
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assert primitive_element(
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[sqrt(2)], x, ex=True, polys=True) == (Poly(x**2 - 2, domain='QQ'), [1], [[1, 0]])
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assert primitive_element([sqrt(2), sqrt(3)], x, ex=True, polys=True) == \
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(Poly(x**4 - 10*x**2 + 1, domain='QQ'), [1, 1], [[Q(1, 2), 0, -Q(9, 2),
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0], [-Q(1, 2), 0, Q(11, 2), 0]])
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assert primitive_element([sqrt(2)], polys=True) == (Poly(x**2 - 2), [1])
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raises(ValueError, lambda: primitive_element([], x, ex=False))
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raises(ValueError, lambda: primitive_element([], x, ex=True))
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# Issue 14117
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a, b = I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3)
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assert primitive_element([a, b, I], x) == (x**4 + 6*x**2 + 1, [1, 0, 0])
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def test_to_number_field():
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assert to_number_field(sqrt(2)) == AlgebraicNumber(sqrt(2))
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assert to_number_field(
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[sqrt(2), sqrt(3)]) == AlgebraicNumber(sqrt(2) + sqrt(3))
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a = AlgebraicNumber(sqrt(2) + sqrt(3), [S.Half, S.Zero, Rational(-9, 2), S.Zero])
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assert to_number_field(sqrt(2), sqrt(2) + sqrt(3)) == a
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assert to_number_field(sqrt(2), AlgebraicNumber(sqrt(2) + sqrt(3))) == a
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raises(IsomorphismFailed, lambda: to_number_field(sqrt(2), sqrt(3)))
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def test_issue_22561():
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a = to_number_field(sqrt(2), sqrt(2) + sqrt(3))
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b = to_number_field(sqrt(2), sqrt(2) + sqrt(5))
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assert field_isomorphism(a, b) == [1, 0]
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def test_issue_22736():
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a = CRootOf(x**4 + x**3 + x**2 + x + 1, -1)
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a._reset()
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b = exp(2*I*pi/5)
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assert field_isomorphism(a, b) == [1, 0]
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