r"""
Functions in ``polys.numberfields.subfield`` solve the "Subfield Problem" and
allied problems, for algebraic number fields.
Following Cohen (see [Cohen93]_ Section 4.5), we can define the main problem as
follows:
* **Subfield Problem:**
Given two number fields $\mathbb{Q}(\alpha)$, $\mathbb{Q}(\beta)$
via the minimal polynomials for their generators $\alpha$ and $\beta$, decide
whether one field is isomorphic to a subfield of the other.
From a solution to this problem flow solutions to the following problems as
well:
* **Primitive Element Problem:**
Given several algebraic numbers
$\alpha_1, \ldots, \alpha_m$, compute a single algebraic number $\theta$
such that $\mathbb{Q}(\alpha_1, \ldots, \alpha_m) = \mathbb{Q}(\theta)$.
* **Field Isomorphism Problem:**
Decide whether two number fields
$\mathbb{Q}(\alpha)$, $\mathbb{Q}(\beta)$ are isomorphic.
* **Field Membership Problem:**
Given two algebraic numbers $\alpha$,
$\beta$, decide whether $\alpha \in \mathbb{Q}(\beta)$, and if so write
$\alpha = f(\beta)$ for some $f(x) \in \mathbb{Q}[x]$.
"""
from sympy.core.add import Add
from sympy.core.numbers import AlgebraicNumber
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify, _sympify
from sympy.ntheory import sieve
from sympy.polys.densetools import dup_eval
from sympy.polys.domains import QQ
from sympy.polys.numberfields.minpoly import _choose_factor, minimal_polynomial
from sympy.polys.polyerrors import IsomorphismFailed
from sympy.polys.polytools import Poly, PurePoly, factor_list
from sympy.utilities import public
from mpmath import MPContext
def is_isomorphism_possible(a, b):
"""Necessary but not sufficient test for isomorphism. """
n = a.minpoly.degree()
m = b.minpoly.degree()
if m % n != 0:
return False
if n == m:
return True
da = a.minpoly.discriminant()
db = b.minpoly.discriminant()
i, k, half = 1, m//n, db//2
while True:
p = sieve[i]
P = p**k
if P > half:
break
if ((da % p) % 2) and not (db % P):
return False
i += 1
return True
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not a.root.is_real or not b.root.is_real:
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
n, m, prev = 100, b.minpoly.degree(), None
ctx = MPContext()
for i in range(1, 5):
A = a.root.evalf(n)
B = b.root.evalf(n)
basis = [1, B] + [ B**i for i in range(2, m) ] + [-A]
ctx.dps = n
coeffs = ctx.pslq(basis, maxcoeff=10**10, maxsteps=1000)
if coeffs is None:
# PSLQ can't find an integer linear combination. Give up.
break
if coeffs != prev:
prev = coeffs
else:
# Increasing precision didn't produce anything new. Give up.
break
# We have
# c0 + c1*B + c2*B^2 + ... + cm-1*B^(m-1) - cm*A ~ 0.
# So bring cm*A to the other side, and divide through by cm,
# for an approximate representation of A as a polynomial in B.
# (We know cm != 0 since `b.minpoly` is irreducible.)
coeffs = [S(c)/coeffs[-1] for c in coeffs[:-1]]
# Throw away leading zeros.
while not coeffs[-1]:
coeffs.pop()
coeffs = list(reversed(coeffs))
h = Poly(coeffs, f.gen, domain='QQ')
# We only have A ~ h(B). We must check whether the relation is exact.
if f.compose(h).rem(g).is_zero:
# Now we know that h(b) is in fact equal to _some conjugate of_ a.
# But from the very precise approximation A ~ h(B) we can assume
# the conjugate is a itself.
return coeffs
else:
n *= 2
return None
def field_isomorphism_factor(a, b):
"""Construct field isomorphism via factorization. """
_, factors = factor_list(a.minpoly, extension=b)
for f, _ in factors:
if f.degree() == 1:
# Any linear factor f(x) represents some conjugate of a in QQ(b).
# We want to know whether this linear factor represents a itself.
# Let f = x - c
c = -f.rep.TC()
# Write c as polynomial in b
coeffs = c.to_sympy_list()
d, terms = len(coeffs) - 1, []
for i, coeff in enumerate(coeffs):
terms.append(coeff*b.root**(d - i))
r = Add(*terms)
# Check whether we got the number a
if a.minpoly.same_root(r, a):
return coeffs
# If none of the linear factors represented a in QQ(b), then in fact a is
# not an element of QQ(b).
return None
@public
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 _switch_domain(g, K):
# An algebraic relation f(a, b) = 0 over Q can also be written
# g(b) = 0 where g is in Q(a)[x] and h(a) = 0 where h is in Q(b)[x].
# This function transforms g into h where Q(b) = K.
frep = g.rep.inject()
hrep = frep.eject(K, front=True)
return g.new(hrep, g.gens[0])
def _linsolve(p):
# Compute root of linear polynomial.
c, d = p.rep.rep
return -d/c
@public
def primitive_element(extension, x=None, *, ex=False, polys=False):
r"""
Find a single generator for a number field given by several generators.
Explanation
===========
The basic problem is this: Given several algebraic numbers
$\alpha_1, \alpha_2, \ldots, \alpha_n$, find a single algebraic number
$\theta$ such that
$\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_n) = \mathbb{Q}(\theta)$.
This function actually guarantees that $\theta$ will be a linear
combination of the $\alpha_i$, with non-negative integer coefficients.
Furthermore, if desired, this function will tell you how to express each
$\alpha_i$ as a $\mathbb{Q}$-linear combination of the powers of $\theta$.
Examples
========
>>> from sympy import primitive_element, sqrt, S, minpoly, simplify
>>> from sympy.abc import x
>>> f, lincomb, reps = primitive_element([sqrt(2), sqrt(3)], x, ex=True)
Then ``lincomb`` tells us the primitive element as a linear combination of
the given generators ``sqrt(2)`` and ``sqrt(3)``.
>>> print(lincomb)
[1, 1]
This means the primtiive element is $\sqrt{2} + \sqrt{3}$.
Meanwhile ``f`` is the minimal polynomial for this primitive element.
>>> print(f)
x**4 - 10*x**2 + 1
>>> print(minpoly(sqrt(2) + sqrt(3), x))
x**4 - 10*x**2 + 1
Finally, ``reps`` (which was returned only because we set keyword arg
``ex=True``) tells us how to recover each of the generators $\sqrt{2}$ and
$\sqrt{3}$ as $\mathbb{Q}$-linear combinations of the powers of the
primitive element $\sqrt{2} + \sqrt{3}$.
>>> print([S(r) for r in reps[0]])
[1/2, 0, -9/2, 0]
>>> theta = sqrt(2) + sqrt(3)
>>> print(simplify(theta**3/2 - 9*theta/2))
sqrt(2)
>>> print([S(r) for r in reps[1]])
[-1/2, 0, 11/2, 0]
>>> print(simplify(-theta**3/2 + 11*theta/2))
sqrt(3)
Parameters
==========
extension : list of :py:class:`~.Expr`
Each expression must represent an algebraic number $\alpha_i$.
x : :py:class:`~.Symbol`, optional (default=None)
The desired symbol to appear in the computed minimal polynomial for the
primitive element $\theta$. If ``None``, we use a dummy symbol.
ex : boolean, optional (default=False)
If and only if ``True``, compute the representation of each $\alpha_i$
as a $\mathbb{Q}$-linear combination over the powers of $\theta$.
polys : boolean, optional (default=False)
If ``True``, return the minimal polynomial as a :py:class:`~.Poly`.
Otherwise return it as an :py:class:`~.Expr`.
Returns
=======
Pair (f, coeffs) or triple (f, coeffs, reps), where:
``f`` is the minimal polynomial for the primitive element.
``coeffs`` gives the primitive element as a linear combination of the
given generators.
``reps`` is present if and only if argument ``ex=True`` was passed,
and is a list of lists of rational numbers. Each list gives the
coefficients of falling powers of the primitive element, to recover
one of the original, given generators.
"""
if not extension:
raise ValueError("Cannot compute primitive element for empty extension")
extension = [_sympify(ext) for ext in extension]
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not ex:
gen, coeffs = extension[0], [1]
g = minimal_polynomial(gen, x, polys=True)
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0)
continue
_, factors = factor_list(g, extension=ext)
g = _choose_factor(factors, x, gen)
s, _, g = g.sqf_norm()
gen += s*ext
coeffs.append(s)
if not polys:
return g.as_expr(), coeffs
else:
return cls(g), coeffs
gen, coeffs = extension[0], [1]
f = minimal_polynomial(gen, x, polys=True)
K = QQ.algebraic_field((f, gen)) # incrementally constructed field
reps = [K.unit] # representations of extension elements in K
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0) # rational ext is not included in the expression of a primitive element
reps.append(K.convert(ext)) # but it is included in reps
continue
p = minimal_polynomial(ext, x, polys=True)
L = QQ.algebraic_field((p, ext))
_, factors = factor_list(f, domain=L)
f = _choose_factor(factors, x, gen)
s, g, f = f.sqf_norm()
gen += s*ext
coeffs.append(s)
K = QQ.algebraic_field((f, gen))
h = _switch_domain(g, K)
erep = _linsolve(h.gcd(p)) # ext as element of K
ogen = K.unit - s*erep # old gen as element of K
reps = [dup_eval(_.rep, ogen, K) for _ in reps] + [erep]
if K.ext.root.is_Rational: # all extensions are rational
H = [K.convert(_).rep for _ in extension]
coeffs = [0]*len(extension)
f = cls(x, domain=QQ)
else:
H = [_.rep for _ in reps]
if not polys:
return f.as_expr(), coeffs, H
else:
return f, coeffs, H
@public
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))