def to_number_field(extension, theta=None, **args):
"""Express `extension` in the field generated by `theta`. """
gen = args.get('gen')
if hasattr(extension, '__iter__'):
extension = list(extension)
else:
extension = [extension]
if len(extension) == 1 and type(extension[0]) is tuple:
return AlgebraicNumber(extension[0])
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))
else:
theta = sympify(theta)
if not theta.is_AlgebraicNumber:
theta = AlgebraicNumber(theta, gen=gen)
coeffs = field_isomorphism(root, theta)
if coeffs is not None:
return AlgebraicNumber(theta, coeffs)
else:
raise IsomorphismFailed("%s is not in a subfield of %s" %
(root, theta.root))
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_pickling_polys_errors():
from sympy.polys.polyerrors import (
ExactQuotientFailed, OperationNotSupported, HeuristicGCDFailed,
HomomorphismFailed, IsomorphismFailed, ExtraneousFactors,
EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible,
NotReversible, NotAlgebraic, DomainError, PolynomialError,
UnificationFailed, GeneratorsError, GeneratorsNeeded,
ComputationFailed, UnivariatePolynomialError,
MultivariatePolynomialError, PolificationFailed, OptionError,
FlagError)
x = Symbol('x')
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (ExactQuotientFailed, ExactQuotientFailed(x, 3*x, ZZ)):
# check(c)
# TODO: TypeError: can't pickle instancemethod objects
# for c in (OperationNotSupported, OperationNotSupported(Poly(x), Poly.gcd)):
# check(c)
for c in (HeuristicGCDFailed, HeuristicGCDFailed()):
check(c)
for c in (HomomorphismFailed, HomomorphismFailed()):
check(c)
for c in (IsomorphismFailed, IsomorphismFailed()):
check(c)
for c in (ExtraneousFactors, ExtraneousFactors()):
check(c)
for c in (EvaluationFailed, EvaluationFailed()):
check(c)
for c in (RefinementFailed, RefinementFailed()):
check(c)
for c in (CoercionFailed, CoercionFailed()):
check(c)
for c in (NotInvertible, NotInvertible()):
check(c)
for c in (NotReversible, NotReversible()):
check(c)
for c in (NotAlgebraic, NotAlgebraic()):
check(c)
for c in (DomainError, DomainError()):
check(c)
for c in (PolynomialError, PolynomialError()):
check(c)
for c in (UnificationFailed, UnificationFailed()):
check(c)
for c in (GeneratorsError, GeneratorsError()):
check(c)
for c in (GeneratorsNeeded, GeneratorsNeeded()):
check(c)
# TODO: PicklingError: Can't pickle <function <lambda> at 0x38578c0>: it's not found as __main__.<lambda>
# for c in (ComputationFailed, ComputationFailed(lambda t: t, 3, None)):
# check(c)
for c in (UnivariatePolynomialError, UnivariatePolynomialError()):
check(c)
for c in (MultivariatePolynomialError, MultivariatePolynomialError()):
check(c)
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (PolificationFailed, PolificationFailed({}, x, x, False)):
# check(c)
for c in (OptionError, OptionError()):
check(c)
for c in (FlagError, FlagError()):
check(c)
