def test_field_isomorphism_pslq():
a = AlgebraicNumber(I)
b = AlgebraicNumber(I * sqrt(3))
pytest.raises(NotImplementedError, lambda: field_isomorphism_pslq(a, b))
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
d = AlgebraicNumber(sqrt(2) + sqrt(3))
e = AlgebraicNumber(sqrt(2) + sqrt(3) + sqrt(7))
assert field_isomorphism_pslq(a, a) == [1, 0]
assert field_isomorphism_pslq(a, b) is None
assert field_isomorphism_pslq(a, c) is None
assert field_isomorphism_pslq(a, d) == [Q(1, 2), 0, -Q(9, 2), 0]
assert field_isomorphism_pslq(
a, e) == [Q(1, 80), 0, -Q(1, 2), 0,
Q(59, 20), 0]
assert field_isomorphism_pslq(b, a) is None
assert field_isomorphism_pslq(b, b) == [1, 0]
assert field_isomorphism_pslq(b, c) is None
assert field_isomorphism_pslq(b, d) == [-Q(1, 2), 0, Q(11, 2), 0]
assert field_isomorphism_pslq(b, e) == [
-Q(3, 640), 0,
Q(67, 320), 0, -Q(297, 160), 0,
Q(313, 80), 0
]
assert field_isomorphism_pslq(c, a) is None
assert field_isomorphism_pslq(c, b) is None
assert field_isomorphism_pslq(c, c) == [1, 0]
assert field_isomorphism_pslq(c, d) is None
assert field_isomorphism_pslq(c, e) == [
Q(3, 640), 0, -Q(71, 320), 0,
Q(377, 160), 0, -Q(469, 80), 0
]
assert field_isomorphism_pslq(d, a) is None
assert field_isomorphism_pslq(d, b) is None
assert field_isomorphism_pslq(d, c) is None
assert field_isomorphism_pslq(d, d) == [1, 0]
assert field_isomorphism_pslq(d, e) == [
-Q(3, 640), 0,
Q(71, 320), 0, -Q(377, 160), 0,
Q(549, 80), 0
]
assert field_isomorphism_pslq(e, a) is None
assert field_isomorphism_pslq(e, b) is None
assert field_isomorphism_pslq(e, c) is None
assert field_isomorphism_pslq(e, d) is None
assert field_isomorphism_pslq(e, e) == [1, 0]
f = AlgebraicNumber(3 * sqrt(2) + 8 * sqrt(7) - 5)
assert field_isomorphism_pslq(
f, e) == [Q(3, 80), 0, -Q(139, 80), 0,
Q(347, 20), 0, -Q(761, 20), -5]
def test_to_number_field():
assert to_number_field(sqrt(2)) == AlgebraicNumber(sqrt(2))
assert to_number_field([sqrt(2),
sqrt(3)]) == AlgebraicNumber(sqrt(2) + sqrt(3))
a = AlgebraicNumber(
sqrt(2) + sqrt(3),
[Rational(1, 2),
Integer(0), -Rational(9, 2),
Integer(0)])
assert to_number_field(sqrt(2), sqrt(2) + sqrt(3)) == a
assert to_number_field(sqrt(2), AlgebraicNumber(sqrt(2) + sqrt(3))) == a
pytest.raises(IsomorphismFailed, lambda: to_number_field(sqrt(2), sqrt(3)))
def test_to_algebraic_integer():
a = AlgebraicNumber(sqrt(3), gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 3
assert a.root == sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(2 * sqrt(3), gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(3) / 2, gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(3) / 2, [Rational(7, 19), 3],
gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(7, 19), QQ(3)], QQ)
def test_to_number_field():
assert to_number_field(sqrt(2)) == AlgebraicNumber(sqrt(2))
assert to_number_field([sqrt(2),
sqrt(3)]) == AlgebraicNumber(sqrt(2) + sqrt(3))
a = AlgebraicNumber(
sqrt(2) + sqrt(3),
[Rational(1, 2),
Integer(0), -Rational(9, 2),
Integer(0)])
assert to_number_field(sqrt(2), sqrt(2) + sqrt(3)) == a
assert to_number_field(sqrt(2), AlgebraicNumber(sqrt(2) + sqrt(3))) == a
pytest.raises(IsomorphismFailed, lambda: to_number_field(sqrt(2), sqrt(3)))
# issue sympy/sympy#5649
assert AlgebraicNumber(1).rep == to_number_field(1, AlgebraicNumber(1)).rep
assert AlgebraicNumber(sqrt(2)).rep == to_number_field(
sqrt(2), AlgebraicNumber(sqrt(2))).rep
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(8, 3)]).rep == DMP([QQ(8, 3)],
QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2),
[Rational(7, 9), Rational(3, 2)]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([-QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple())
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))
def test_field_isomorphism():
assert field_isomorphism(3, sqrt(2)) == [3]
assert field_isomorphism(I * sqrt(3), I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(-I * sqrt(3), I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(I * sqrt(3), -I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(-I * sqrt(3), -I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(2 * I * sqrt(3) / 7,
5 * I * sqrt(3) / 3) == [Rational(6, 35), 0]
assert field_isomorphism(-2 * I * sqrt(3) / 7,
5 * I * sqrt(3) / 3) == [-Rational(6, 35), 0]
assert field_isomorphism(2 * I * sqrt(3) / 7,
-5 * I * sqrt(3) / 3) == [-Rational(6, 35), 0]
assert field_isomorphism(-2 * I * sqrt(3) / 7,
-5 * I * sqrt(3) / 3) == [Rational(6, 35), 0]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [Rational(6, 35), 27]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [-Rational(6, 35), 27]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [-Rational(6, 35), 27]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [Rational(6, 35), 27]
p = AlgebraicNumber(sqrt(2) + sqrt(3))
q = AlgebraicNumber(-sqrt(2) + sqrt(3))
r = AlgebraicNumber(sqrt(2) - sqrt(3))
s = AlgebraicNumber(-sqrt(2) - sqrt(3))
pos_coeffs = [Rational(1, 2), Integer(0), -Rational(9, 2), Integer(0)]
neg_coeffs = [-Rational(1, 2), Integer(0), Rational(9, 2), Integer(0)]
a = AlgebraicNumber(sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = AlgebraicNumber(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [Rational(1, 2), Integer(0), -Rational(11, 2), Integer(0)]
neg_coeffs = [-Rational(1, 2), Integer(0), Rational(11, 2), Integer(0)]
a = AlgebraicNumber(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [Rational(3, 2), Integer(0), -Rational(33, 2), -Integer(8)]
neg_coeffs = [-Rational(3, 2), Integer(0), Rational(33, 2), -Integer(8)]
a = AlgebraicNumber(3 * sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(3 * sqrt(2) + 2 * sqrt(3) + 1)
pos_1_coeffs = [Rational(1, 2), Integer(0), -Rational(5, 2), Integer(1)]
neg_5_coeffs = [-Rational(5, 2), Integer(0), Rational(49, 2), Integer(1)]
pos_5_coeffs = [Rational(5, 2), Integer(0), -Rational(49, 2), Integer(1)]
neg_1_coeffs = [-Rational(1, 2), Integer(0), Rational(5, 2), Integer(1)]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
# issue 5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3)) +
2 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3))))
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex, x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex, x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = (expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))**Rational(1, 3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(
-sqrt(5) / 2 - S.Half + (-sqrt(5) / 2 - S.Half)**2, x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3, x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I, x,
compose=False) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1, x,
compose=False) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
x,
compose=False) == x**4 + 18 * x**2 + 49
def test_pickling_polys_numberfields():
for c in (AlgebraicNumber, AlgebraicNumber(sqrt(3))):
check(c)
def to_diofant(self, a):
"""Convert ``a`` to a Diofant object. """
from diofant.polys.numberfields import AlgebraicNumber
return AlgebraicNumber(self.ext, a).as_expr()
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(8, 3)]).rep == DMP([QQ(8, 3)],
QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2),
[Rational(7, 9), Rational(3, 2)]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple(1, 0))
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(1, 0), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(2, 5))
pytest.raises(ValueError,
lambda: AlgebraicNumber(RootOf(x**3 + y * x + 1, x, 0)))
a = AlgebraicNumber(RootOf(x**3 + 2 * x - 1, x, 1), alias='alpha')
assert a.free_symbols == set()
# integer powers:
assert a**0 == 1
assert a**2 == AlgebraicNumber(a, (1, 0, 0), alias='alpha')
assert a**5 == AlgebraicNumber(a, (1, 0, 0, 0, 0, 0), alias='alpha')
assert a**110 == AlgebraicNumber(a, ([1] + [0] * 110), alias='alpha')
assert (a**pi).is_Pow
b = AlgebraicNumber(sqrt(2), (1, 0), alias='theta')
c = b + 1
assert c**2 == 2 * b + 3
assert c**5 == 29 * b + 41
assert c**-2 == 3 - 2 * b
assert c**-11 == 5741 * b - 8119
# arithmetics
assert a**3 == -2 * a + 1 == a * (-2) + 1 == 1 + (-2) * a == 1 - 2 * a
assert a**5 == a**2 + 4 * a - 2
assert a**4 == -2 * a**2 + a == a - 2 * a**2
assert a**110 == (-2489094528619081871 * a**2 + 3737645722703173544 * a -
1182958048412500088)
assert a + a == 2 * a
assert 2 * a - a == a
assert Integer(1) - a == (-a) + 1
assert (a + pi).is_Add
assert (pi + a).is_Add
assert (a - pi).is_Add
assert (pi - a).is_Add
assert (a * pi).is_Mul
assert (pi * a).is_Mul
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(Rational(1, 3), x, compose=False) == 3 * x - 1
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(pi, x, compose=False))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(sin(sqrt(2)), x, compose=False))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(2**pi, x, compose=False))
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
e = 1 / sqrt(sqrt(1 + sqrt(3)) - 4)
assert minimal_polynomial(e, x) == minimal_polynomial(e, x, compose=False)
assert minimal_polynomial(e, x) == (222 * x**8 + 240 * x**6 + 94 * x**4 +
16 * x**2 + 1)
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a), x, polys=True) == Poly(x**4 - 2)
assert minimal_polynomial(a + 1, x, polys=True) == Poly(x**2 - 2 * x - 1)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
# issue diofant/diofant#431
theta = AlgebraicNumber(sqrt(2), (S.Half, 17))
assert minimal_polynomial(theta, x) == 2 * x**2 - 68 * x + 577
theta = AlgebraicNumber(RootOf(x**7 + x - 1, x, 3), (1, 2, 0, 0, 1))
ans = minimal_polynomial(theta, x)
assert ans == (x**7 - 7 * x**6 + 19 * x**5 - 27 * x**4 + 63 * x**3 -
115 * x**2 + 82 * x - 147)
assert minimal_polynomial(theta.as_expr(), x, compose=False) == ans
theta = AlgebraicNumber(RootOf(x**5 + 5 * x - 1, x, 2), (1, -1, 1))
ans = (x**30 - 15 * x**28 - 10 * x**27 + 135 * x**26 + 330 * x**25 -
705 * x**24 - 150 * x**23 + 3165 * x**22 - 6850 * x**21 +
7182 * x**20 + 3900 * x**19 + 4435 * x**18 + 11970 * x**17 -
259725 * x**16 - 18002 * x**15 + 808215 * x**14 - 200310 * x**13 -
647115 * x**12 + 299280 * x**11 - 1999332 * x**10 + 910120 * x**9 +
2273040 * x**8 - 5560320 * x**7 + 5302000 * x**6 - 2405376 * x**5 +
1016640 * x**4 - 804480 * x**3 + 257280 * x**2 - 53760 * x + 1280)
assert minimal_polynomial(sqrt(theta) + root(theta, 3), x) == ans
theta = sqrt(
1 + 1 /
(AlgebraicNumber(RootOf(x**3 + 4 * x - 15, x, 1), (1, 0, 1)) + 1 /
(sqrt(3) + AlgebraicNumber(RootOf(x**3 - x + 1, x, 0), (1, 2, -1)))))
ans = (2262264837876687263 * x**36 - 38939909597855051866 * x**34 +
315720420314462950715 * x**32 - 1601958657418182606114 * x**30 +
5699493671077371036494 * x**28 - 15096777696140985506150 * x**26 +
30847690820556462893974 * x**24 - 49706549068200640994022 * x**22 +
64013601241426223813103 * x**20 - 66358713088213594372990 * x**18 +
55482571280904904971976 * x**16 - 37309340229165533529076 * x**14 +
20016999328983554519040 * x**12 - 8446273798231518826782 * x**10 +
2738866994867366499481 * x**8 - 657825125060873756424 * x**6 +
110036313740049140508 * x**4 - 11416087328869938298 * x**2 +
551322649782053543)
assert minimal_polynomial(theta, x) == ans
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
a = AlgebraicNumber(RootOf(x**3 + x - 1, x, 0))
assert minimal_polynomial(1 / a**2, x) == x**3 - x**2 - 2 * x - 1
# issue sympy/sympy#5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3)) +
2 *
(Rational(-1, 17280000) + sqrt(15) * I / 28800000)**Rational(1, 3))))
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex, x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex, x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = (expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))**Rational(1, 3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(
-sqrt(5) / 2 - S.Half + (-sqrt(5) / 2 - S.Half)**2, x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3, x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I, x,
compose=False) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1, x,
compose=False) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
x,
compose=False) == x**4 + 18 * x**2 + 49