Is there any way to define a polynomial ring over a sparse quotient
ring?
This works:
sage: p = 5
sage: A.<T> = PolynomialRing(Integers(p))
sage: f = T^2+1
sage: B = A.quo(f)
sage: print B
sage: C.<s> = PolynomialRing(B)
sage: print C
Univariate Quotient Polynomial Ring in Tbar over Ring of integers
modulo 5 with modulus T^2 + 1
Univariate Polynomial Ring in s over Univariate Quotient Polynomial
Ring in Tbar over Ring of integers modulo 5 with modulus T^2 + 1
But this does not work:
sage: p = 5
sage: A.<T> = PolynomialRing(Integers(p),sparse=True)
sage: f = T^2+1
sage: B = A.quo(f)
sage: print B
sage: C.<s> = PolynomialRing(B)
Univariate Quotient Polynomial Ring in Tbar over Ring of integers
modulo 5 with modulus T^2 + 1
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/kvankov/.sage/sage_notebook/worksheets/admin/16/code/
189.py", line 11, in <module>
exec compile(ur'C = PolynomialRing(B,names=(\u0027s\u0027,)); (s,)
= C._first_ngens(Integer(1))' + '\n', '', 'single')
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sympy/plotting/", line 1, in <module>
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/polynomial/
polynomial_ring_constructor.py", line 256, in PolynomialRing
R = _single_variate(base_ring, name, sparse)
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/polynomial/
polynomial_ring_constructor.py", line 329, in _single_variate
elif base_ring.is_field():
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/polynomial/
polynomial_quotient_ring.py", line 412, in is_field
return self.base_ring().is_field() and
self.modulus().is_irreducible()
File "polynomial_element.pyx", line 3762, in
sage.rings.polynomial.polynomial_element.Polynomial.is_irreducible
File "polynomial_element.pyx", line 2109, in
sage.rings.polynomial.polynomial_element.Polynomial.factor
File "polynomial_element.pyx", line 2118, in
sage.rings.polynomial.polynomial_element.Polynomial._factor_pari_helper
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/polynomial/polynomial_ring.py",
line 243, in __call__
return C(self, x, check, is_gen, construct=construct)
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/polynomial/
polynomial_element_generic.py", line 95, in __init__
self.__coeffs[i] = R(z)
File "/usr/local/sage-2.11-ubuntu32-intel-i686-Linux/local/lib/
python2.5/site-packages/sage/rings/integer_mod_ring.py", line 579, in
__call__
return integer_mod.IntegerMod(self, x)
File "integer_mod.pyx", line 129, in
sage.rings.integer_mod.IntegerMod
File "integer_mod.pyx", line 1380, in
sage.rings.integer_mod.IntegerMod_int.__init__
File "integer_ring.pyx", line 274, in
sage.rings.integer_ring.IntegerRing_class.__call__
File "integer.pyx", line 377, in sage.rings.integer.Integer.__init__
File "gen.pyx", line 750, in sage.libs.pari.gen.gen.__hex__
TypeError: gen must be of PARI type t_INT
K.
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