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 == [alpha.field_element([1]),
alpha.field_element([S.Half, S.Half])]
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), [S.Half, S.Zero, Rational(-9, 2), S.Zero])
assert to_number_field(sqrt(2), sqrt(2) + sqrt(3)) == a
assert to_number_field(sqrt(2), AlgebraicNumber(sqrt(2) + sqrt(3))) == a
raises(IsomorphismFailed, lambda: to_number_field(sqrt(2), sqrt(3)))
def test_AlgebraicNumber():
a = AlgebraicNumber(sqrt(2))
sT(
a,
"AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])"
)
a = AlgebraicNumber(root(-2, 3))
sT(
a,
"AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])"
)
def test_evalf_real_alg_num():
# This test demonstrates why the entry for `AlgebraicNumber` in
# `sympy.core.evalf._create_evalf_table()` has to use `x.to_root()`,
# instead of `x.as_expr()`. If the latter is used, then `z` will be
# a complex number with `0.e-20` for imaginary part, even though `a5`
# is a real number.
zeta = Symbol('zeta')
a5 = AlgebraicNumber(CRootOf(cyclotomic_poly(5), -1), [-1, -1, 0, 0],
alias=zeta)
z = a5.evalf()
assert isinstance(z, Float)
assert not hasattr(z, '_mpc_')
assert hasattr(z, '_mpf_')
def test_field_isomorphism_pslq():
a = AlgebraicNumber(I)
b = AlgebraicNumber(I * sqrt(3))
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_NumberKind():
assert S.One.kind is NumberKind
assert pi.kind is NumberKind
assert S.NaN.kind is NumberKind
assert zoo.kind is NumberKind
assert I.kind is NumberKind
assert AlgebraicNumber(1).kind is NumberKind
def test_AlgebraicField_alias():
# No default alias:
k = QQ.algebraic_field(sqrt(2))
assert k.ext.alias is None
# For a single extension, its alias is used:
alpha = AlgebraicNumber(sqrt(2), alias='alpha')
k = QQ.algebraic_field(alpha)
assert k.ext.alias.name == 'alpha'
# Can override the alias of a single extension:
k = QQ.algebraic_field(alpha, alias='theta')
assert k.ext.alias.name == 'theta'
# With multiple extensions, no default alias:
k = QQ.algebraic_field(sqrt(2), sqrt(3))
assert k.ext.alias is None
# With multiple extensions, no default alias, even if one of
# the extensions has one:
k = QQ.algebraic_field(alpha, sqrt(3))
assert k.ext.alias is None
# With multiple extensions, may set an alias:
k = QQ.algebraic_field(sqrt(2), sqrt(3), alias='theta')
assert k.ext.alias.name == 'theta'
# Alias is passed to constructed field elements:
k = QQ.algebraic_field(alpha)
beta = k.to_alg_num(k([1, 2, 3]))
assert beta.alias is alpha.alias
def test_AlgebraicNumber_to_root():
assert AlgebraicNumber(sqrt(2)).to_root() == sqrt(2)
zeta5_squared = AlgebraicNumber(CRootOf(x**5 - 1, 4), coeffs=[1, 0, 0])
assert zeta5_squared.to_root() == CRootOf(x**4 + x**3 + x**2 + x + 1, 1)
zeta3_squared = AlgebraicNumber(CRootOf(x**3 - 1, 2), coeffs=[1, 0, 0])
assert zeta3_squared.to_root() == -S(1)/2 - sqrt(3)*I/2
assert zeta3_squared.to_root(radicals=False) == CRootOf(x**2 + x + 1, 0)
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 alg_field_from_poly(self, poly, alias=None, root_index=-1):
r"""
Convenience method to construct an algebraic extension on a root of a
polynomial, chosen by root index.
Parameters
==========
poly : :py:class:`~.Poly`
The polynomial whose root generates the extension.
alias : str, optional (default=None)
Symbol name for the generator of the extension.
E.g. "alpha" or "theta".
root_index : int, optional (default=-1)
Specifies which root of the polynomial is desired. The ordering is
as defined by the :py:class:`~.ComplexRootOf` class. The default of
``-1`` selects the most natural choice in the common cases of
quadratic and cyclotomic fields (the square root on the positive
real or imaginary axis, resp. $\mathrm{e}^{2\pi i/n}$).
Examples
========
>>> from sympy import QQ, Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 - 2)
>>> K = QQ.alg_field_from_poly(f)
>>> K.ext.minpoly == f
True
>>> g = Poly(8*x**3 - 6*x - 1)
>>> L = QQ.alg_field_from_poly(g, "alpha")
>>> L.ext.minpoly == g
True
>>> L.to_sympy(L([1, 1, 1]))
alpha**2 + alpha + 1
"""
from sympy.polys.rootoftools import CRootOf
root = CRootOf(poly, root_index)
alpha = AlgebraicNumber(root, alias=alias)
return self.algebraic_field(alpha)
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 = [ S.Half, S.Zero, Rational(-9, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(9, 2), S.Zero]
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 = [ S.Half, S.Zero, Rational(-11, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(11, 2), S.Zero]
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), S.Zero, Rational(-33, 2), -S(8)]
neg_coeffs = [Rational(-3, 2), S.Zero, Rational(33, 2), -S(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 = [ S.Half, S.Zero, Rational(-5, 2), S.One]
neg_5_coeffs = [Rational(-5, 2), S.Zero, Rational(49, 2), S.One]
pos_5_coeffs = [ Rational(5, 2), S.Zero, Rational(-49, 2), S.One]
neg_1_coeffs = [Rational(-1, 2), S.Zero, Rational(5, 2), S.One]
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
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(2 ** (S(1) / 3))
assert is_isomorphism_possible(a, b) is False
assert field_isomorphism(a, b) is None
def test_core_numbers():
for c in (Integer(2), Rational(2, 3), Float("1.2")):
check(c)
for c in (AlgebraicNumber, AlgebraicNumber(sqrt(3))):
check(c, check_attr=False)
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
raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
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,
domain='QQ')
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2, domain='QQ')
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, domain='QQ')
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3, domain='QQ')
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 = S('''
-1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)))''')
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = (sqrt(5) * sqrt(I) / (5 * sqrt(1 + 125 * I)) + 25 * sqrt(5) /
(I**Q(5, 2) * (1 + 125 * I)**Q(3, 2)) + 3125 * sqrt(5) /
(I**Q(11, 2) * (1 + 125 * I)**Q(3, 2)) + 5 * I * sqrt(1 - I / 125))
mp = minimal_polynomial(ex, x)
assert mp == 25 * x**4 + 5000 * x**2 + 250016
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(S("-sqrt(5)/2 - 1/2 + (-sqrt(5)/2 - 1/2)**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
# minimal polynomial of I
assert minimal_polynomial(I, x, domain=QQ.algebraic_field(I)) == x - I
K = QQ.algebraic_field(I * (sqrt(2) + 1))
assert minimal_polynomial(I, x, domain=K) == x - I
assert minimal_polynomial(I, x, domain=QQ) == x**2 + 1
assert minimal_polynomial(I, x, domain='QQ(y)') == x**2 + 1
#issue 11553
assert minimal_polynomial(GoldenRatio, x) == x**2 - x - 1
assert minimal_polynomial(TribonacciConstant + 3,
x) == x**3 - 10 * x**2 + 32 * x - 34
assert minimal_polynomial(GoldenRatio, x, domain=QQ.algebraic_field(sqrt(5))) == \
2*x - sqrt(5) - 1
assert minimal_polynomial(TribonacciConstant, x, domain=QQ.algebraic_field(cbrt(19 - 3*sqrt(33)))) == \
48*x - 19*(19 - 3*sqrt(33))**Rational(2, 3) - 3*sqrt(33)*(19 - 3*sqrt(33))**Rational(2, 3) \
- 16*(19 - 3*sqrt(33))**Rational(1, 3) - 16
# AlgebraicNumber with an alias.
# Wester H24
phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
assert minimal_polynomial(phi, x) == x**2 - x - 1
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() == [S.One, S.Zero]
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), ()).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), (0, 0)).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() == [S.One, S(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)
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, domain='QQ')
assert b.as_poly() == Poly(y, domain='QQ')
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, domain='QQ')
assert b.as_poly() == Poly(2*y + 3, domain='QQ')
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(1, 2, 3))
a = AlgebraicNumber(sqrt(2), [1, 2], "alpha")
b = AlgebraicNumber(a)
c = AlgebraicNumber(a, alias="gamma")
assert a == b
assert c.alias.name == "gamma"
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1)/2, 0, S(-9)/2, 0])
b = AlgebraicNumber(a, [1, 0, 0])
assert b.root == a.root
assert a.to_root() == sqrt(2)
assert b.to_root() == 2
a = AlgebraicNumber(2)
assert a.is_primitive_element is True
def field_isomorphism(a, b, *, fast=True):
r"""
Find an embedding of one number field into another.
Explanation
===========
This function looks for an isomorphism from $\mathbb{Q}(a)$ onto some
subfield of $\mathbb{Q}(b)$. Thus, it solves the Subfield Problem.
Examples
========
>>> from sympy import sqrt, field_isomorphism, I
>>> print(field_isomorphism(3, sqrt(2))) # doctest: +SKIP
[3]
>>> print(field_isomorphism( I*sqrt(3), I*sqrt(3)/2)) # doctest: +SKIP
[2, 0]
Parameters
==========
a : :py:class:`~.Expr`
Any expression representing an algebraic number.
b : :py:class:`~.Expr`
Any expression representing an algebraic number.
fast : boolean, optional (default=True)
If ``True``, we first attempt a potentially faster way of computing the
isomorphism, falling back on a slower method if this fails. If
``False``, we go directly to the slower method, which is guaranteed to
return a result.
Returns
=======
List of rational numbers, or None
If $\mathbb{Q}(a)$ is not isomorphic to some subfield of
$\mathbb{Q}(b)$, then return ``None``. Otherwise, return a list of
rational numbers representing an element of $\mathbb{Q}(b)$ to which
$a$ may be mapped, in order to define a monomorphism, i.e. an
isomorphism from $\mathbb{Q}(a)$ to some subfield of $\mathbb{Q}(b)$.
The elements of the list are the coefficients of falling powers of $b$.
"""
a, b = sympify(a), sympify(b)
if not a.is_AlgebraicNumber:
a = AlgebraicNumber(a)
if not b.is_AlgebraicNumber:
b = AlgebraicNumber(b)
a = a.to_primitive_element()
b = b.to_primitive_element()
if a == b:
return a.coeffs()
n = a.minpoly.degree()
m = b.minpoly.degree()
if n == 1:
return [a.root]
if m % n != 0:
return None
if fast:
try:
result = field_isomorphism_pslq(a, b)
if result is not None:
return result
except NotImplementedError:
pass
return field_isomorphism_factor(a, b)
def test_issue_22561():
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1) / 2, 0, S(-9) / 2, 0], gen=x)
assert a.as_expr() == sqrt(2)
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(a**3, x) == x**2 - 8
def test_issue_22559():
alpha = AlgebraicNumber(sqrt(2))
assert minimal_polynomial(alpha**3, x) == x**2 - 8
def to_number_field(extension, theta=None, *, gen=None, alias=None):
r"""
Express one algebraic number in the field generated by another.
Explanation
===========
Given two algebraic numbers $\eta, \theta$, this function either expresses
$\eta$ as an element of $\mathbb{Q}(\theta)$, or else raises an exception
if $\eta \not\in \mathbb{Q}(\theta)$.
This function is essentially just a convenience, utilizing
:py:func:`~.field_isomorphism` (our solution of the Subfield Problem) to
solve this, the Field Membership Problem.
As an additional convenience, this function allows you to pass a list of
algebraic numbers $\alpha_1, \alpha_2, \ldots, \alpha_n$ instead of $\eta$.
It then computes $\eta$ for you, as a solution of the Primitive Element
Problem, using :py:func:`~.primitive_element` on the list of $\alpha_i$.
Examples
========
>>> from sympy import sqrt, to_number_field
>>> eta = sqrt(2)
>>> theta = sqrt(2) + sqrt(3)
>>> a = to_number_field(eta, theta)
>>> print(type(a))
<class 'sympy.core.numbers.AlgebraicNumber'>
>>> a.root
sqrt(2) + sqrt(3)
>>> print(a)
sqrt(2)
>>> a.coeffs()
[1/2, 0, -9/2, 0]
We get an :py:class:`~.AlgebraicNumber`, whose ``.root`` is $\theta$, whose
value is $\eta$, and whose ``.coeffs()`` show how to write $\eta$ as a
$\mathbb{Q}$-linear combination in falling powers of $\theta$.
Parameters
==========
extension : :py:class:`~.Expr` or list of :py:class:`~.Expr`
Either the algebraic number that is to be expressed in the other field,
or else a list of algebraic numbers, a primitive element for which is
to be expressed in the other field.
theta : :py:class:`~.Expr`, None, optional (default=None)
If an :py:class:`~.Expr` representing an algebraic number, behavior is
as described under **Explanation**. If ``None``, then this function
reduces to a shorthand for calling :py:func:`~.primitive_element` on
``extension`` and turning the computed primitive element into an
:py:class:`~.AlgebraicNumber`.
gen : :py:class:`~.Symbol`, None, optional (default=None)
If provided, this will be used as the generator symbol for the minimal
polynomial in the returned :py:class:`~.AlgebraicNumber`.
alias : str, :py:class:`~.Symbol`, None, optional (default=None)
If provided, this will be used as the alias symbol for the returned
:py:class:`~.AlgebraicNumber`.
Returns
=======
AlgebraicNumber
Belonging to $\mathbb{Q}(\theta)$ and equaling $\eta$.
Raises
======
IsomorphismFailed
If $\eta \not\in \mathbb{Q}(\theta)$.
See Also
========
field_isomorphism
primitive_element
"""
if hasattr(extension, '__iter__'):
extension = list(extension)
else:
extension = [extension]
if len(extension) == 1 and isinstance(extension[0], tuple):
return AlgebraicNumber(extension[0], alias=alias)
minpoly, coeffs = primitive_element(extension, gen, polys=True)
root = sum([ coeff*ext for coeff, ext in zip(coeffs, extension) ])
if theta is None:
return AlgebraicNumber((minpoly, root), alias=alias)
else:
theta = sympify(theta)
if not theta.is_AlgebraicNumber:
theta = AlgebraicNumber(theta, gen=gen, alias=alias)
coeffs = field_isomorphism(root, theta)
if coeffs is not None:
return AlgebraicNumber(theta, coeffs, alias=alias)
else:
raise IsomorphismFailed(
"%s is not in a subfield of %s" % (root, theta.root))
def test_issue_22849():
a = -8 + 3 * sqrt(3)
x = AlgebraicNumber(a)
assert evalf(a, 1, {}) == evalf(x, 1, {})