I'm trying to check an ideal in a multivariate polynomial ring over $\overline{\mathbb Q}$ for primality.
Using just a single variable, I can just type
R.<x> = PolynomialRing(QQ.algebraic_closure())
print Ideal([x^2+2]).is_prime()
and Sage tells me that $\langle x^2+2\rangle$ is not prime in $\overline{\mathbb Q}[x]$, as expected.
However, as soon as I add another variable, e.g. R.<x,y> = ... I get the following output:
verbose 0 (3539: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-8-c0542a5be0f8> in <module>()
----> 1 print Ideal([x**Integer(2)+Integer(2)]).is_prime()
/opt/sage-6.4.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in is_prime(self, **kwds)
1265 if not self.ring().base_ring().is_field():
1266 raise NotImplementedError
-> 1267 CPD = self.complete_primary_decomposition(**kwds)
1268 if len(CPD) != 1:
1269 return False
/opt/sage-6.4.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in __call__(self, *args, **kwds)
602 if not R.base_ring().is_field():
603 raise ValueError("Coefficient ring must be a field for function '%s'."%(self.f.__name__))
--> 604 return self.f(self._instance, *args, **kwds)
605
606 require_field = RequireField
/opt/sage-6.4.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in wrapper(*args, **kwds)
503 """
504 with LibSingularDefaultContext():
--> 505 return func(*args, **kwds)
506 return wrapper
507
/opt/sage-6.4.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in complete_primary_decomposition(self, algorithm)
1079 if algorithm == 'sy':
1080 primdecSY = sage.libs.singular.ff.primdec__lib.primdecSY
-> 1081 P = primdecSY(self)
1082 elif algorithm == 'gtz':
1083 primdecGTZ = sage.libs.singular.ff.primdec__lib.primdecGTZ
/opt/sage-6.4.beta4/local/lib/python2.7/site-packages/sage/libs/singular/function.so in sage.libs.singular.function.SingularFunction.__call__ (build/cythonized/sage/libs/singular/function.cpp:13788)()
TypeError: Cannot call Singular function 'primdecSY' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain_with_category'>'
Is there a way to work around this or is there really no implementation for primality checks in $\overline{\mathbb Q}[x_1,\dots,x_n]$?
