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(1)/2, S(0), -S(9)/2, S(0)])
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, "to_number_field(sqrt(2), sqrt(3))")
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 __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
if len(ext) == 1 and isinstance(ext[0], tuple):
orig_ext = ext[0][1:]
else:
orig_ext = ext
self.orig_ext = orig_ext
"""
Original elements given to generate the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.orig_ext
(sqrt(2), sqrt(3))
"""
self.ext = to_number_field(ext)
"""
Primitive element used for the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.ext
sqrt(2) + sqrt(3)
"""
self.mod = self.ext.minpoly.rep
"""
Minimal polynomial for the primitive element of the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2))
>>> K.mod
DMP([1, 0, -2], QQ, None)
"""
self.domain = self.dom = dom
self.ngens = 1
self.symbols = self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
self._maximal_order = None
self._discriminant = None
self._nilradicals_mod_p = {}
def from_sympy(self, a):
"""Convert SymPy's expression to `dtype`. """
try:
return self([self.dom.from_sympy(a)])
except CoercionFailed:
pass
from sympy.polys.numberfields import to_number_field
try:
return self(to_number_field(a, self.ext).native_coeffs())
except (NotAlgebraic, IsomorphismFailed):
raise CoercionFailed("%s is not a valid algebraic number in %s" % (a, self))
def from_sympy(self, a):
"""Convert SymPy's expression to ``dtype``. """
try:
return self([self.dom.from_sympy(a)])
except CoercionFailed:
pass
from sympy.polys.numberfields import to_number_field
try:
return self(to_number_field(a, self.ext).native_coeffs())
except (NotAlgebraic, IsomorphismFailed):
raise CoercionFailed("%s is not a valid algebraic number in %s" % (a, self))
def __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
self.ext = to_number_field(ext)
self.mod = self.ext.minpoly.rep
self.dom = dom
self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
def __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
self.ext = to_number_field(ext)
self.mod = self.ext.minpoly.rep
self.dom = dom
self.gens = (self.ext,)
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
def __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
if len(ext) == 1 and isinstance(ext[0], tuple):
self.orig_ext = ext[0][1:]
else:
self.orig_ext = ext
self.ext = to_number_field(ext)
self.mod = self.ext.minpoly.rep
self.domain = self.dom = dom
self.ngens = 1
self.symbols = self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
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_aliased is False
assert a.coeffs() == [S(1), S(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_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_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [S(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), [S(7) / 9, S(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_aliased is False
assert a.coeffs() == [S(1), 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_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_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_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(1), S(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), ()).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), [S(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), [S(7)/9, S(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(1), 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)
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(1, 2, 3))
def __init__(self, dom, *ext, alias=None):
r"""
Parameters
==========
dom : :py:class:`~.Domain`
The base field over which this is an extension field.
Currently only :ref:`QQ` is accepted.
*ext : One or more :py:class:`~.Expr`
Generators of the extension. These should be expressions that are
algebraic over `\mathbb{Q}`.
alias : str, :py:class:`~.Symbol`, None, optional (default=None)
If provided, this will be used as the alias symbol for the
primitive element of the :py:class:`~.AlgebraicField`.
If ``None``, while ``ext`` consists of exactly one
:py:class:`~.AlgebraicNumber`, its alias (if any) will be used.
"""
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
if len(ext) == 1 and isinstance(ext[0], tuple):
orig_ext = ext[0][1:]
else:
orig_ext = ext
if alias is None and len(ext) == 1:
alias = getattr(ext[0], 'alias', None)
self.orig_ext = orig_ext
"""
Original elements given to generate the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.orig_ext
(sqrt(2), sqrt(3))
"""
self.ext = to_number_field(ext, alias=alias)
"""
Primitive element used for the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.ext
sqrt(2) + sqrt(3)
"""
self.mod = self.ext.minpoly.rep
"""
Minimal polynomial for the primitive element of the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2))
>>> K.mod
DMP([1, 0, -2], QQ, None)
"""
self.domain = self.dom = dom
self.ngens = 1
self.symbols = self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
self._maximal_order = None
self._discriminant = None
self._nilradicals_mod_p = {}
