def _gcd_univariate_polynomial(self, f, g):
"""
Return the greatest common divisor of ``f`` and ``g``, as a
monic polynomial.
INPUT:
- ``f``, ``g`` -- two polynomials defined over ``self``
.. NOTE::
This is a helper method for
:meth:`sage.rings.polynomial.polynomial_element.Polynomial.gcd`.
EXAMPLES::
sage: R.<x> = QQbar[]
sage: QQbar._gcd_univariate_polynomial(2*x,2*x^2)
x
"""
ret = EuclideanDomains().ElementMethods().gcd(f,g)
c = ret.leading_coefficient()
if c.is_unit():
return (1/c)*ret
return ret
def _gcd_univariate_polynomial(self, f, g):
"""
Return the greatest common divisor of ``f`` and ``g``, as a
monic polynomial.
INPUT:
- ``f``, ``g`` -- two polynomials defined over ``self``
.. NOTE::
This is a helper method for
:meth:`sage.rings.polynomial.polynomial_element.Polynomial.gcd`.
EXAMPLES::
sage: R.<x> = QQbar[]
sage: QQbar._gcd_univariate_polynomial(2*x,2*x^2)
x
"""
ret = EuclideanDomains().element_class.gcd(f, g)
c = ret.leading_coefficient()
if c.is_unit():
return (1 / c) * ret
return ret
def gcd(self, other):
"""
Return the greatest common divisor of this polynomial and ``other``, as
a monic polynomial.
INPUT:
- ``other`` -- a polynomial defined over the same ring as ``self``
EXAMPLES::
sage: R.<x> = QQbar[]
sage: (2*x).gcd(2*x^2)
x
sage: zero = R.zero()
sage: zero.gcd(2*x)
x
sage: (2*x).gcd(zero)
x
sage: zero.gcd(zero)
0
"""
from sage.categories.euclidean_domains import EuclideanDomains
g = EuclideanDomains().ElementMethods().gcd(self, other)
c = g.leading_coefficient()
if c.is_unit():
return (1 / c) * g
return g
def gcd(self, other):
"""
Return the greatest common divisor of this polynomial and ``other``, as
a monic polynomial.
INPUT:
- ``other`` -- a polynomial defined over the same ring as ``self``
EXAMPLES::
sage: R.<x> = QQbar[]
sage: (2*x).gcd(2*x^2)
x
sage: zero = R.zero()
sage: zero.gcd(2*x)
x
sage: (2*x).gcd(zero)
x
sage: zero.gcd(zero)
0
"""
from sage.categories.euclidean_domains import EuclideanDomains
g = EuclideanDomains().ElementMethods().gcd(self, other)
c = g.leading_coefficient()
if c.is_unit():
return (1/c)*g
return g
def super_categories(self):
"""
EXAMPLES::
sage: DiscreteValuationRings().super_categories()
[Category of euclidean domains]
"""
return [EuclideanDomains()]
def extra_super_categories(self):
"""
EXAMPLES::
sage: Fields().extra_super_categories()
[Category of euclidean domains]
"""
return [EuclideanDomains()]
def super_categories(self):
"""
EXAMPLES::
sage: Fields().super_categories()
[Category of euclidean domains, Category of unique factorization domains, Category of division rings]
"""
return [EuclideanDomains(), UniqueFactorizationDomains(), DivisionRings()]
def _convert_map_from_(self, R):
"""
Finds conversion maps from R to this ring.
Currently, a conversion exists if the defining polynomial is the same.
EXAMPLES::
sage: R.<a> = Zq(125)
sage: S = R.change(type='capped-abs', prec=40, print_mode='terse', print_pos=False)
sage: S(a - 15)
-15 + a + O(5^20)
We get conversions from the exact field::
sage: K = R.exact_field(); K
Number Field in a with defining polynomial x^3 + 3*x + 3
sage: R(K.gen())
a + O(5^20)
and its maximal order::
sage: OK = K.maximal_order()
sage: R(OK.gen(1))
a + O(5^20)
"""
cat = None
if self._implementation == 'NTL' and R == QQ:
# Want to use DefaultConvertMap_unique
return None
if isinstance(R, pAdicExtensionGeneric) and R.prime() == self.prime(
) and R.defining_polynomial(exact=True) == self.defining_polynomial(
exact=True):
if R.is_field() and not self.is_field():
cat = SetsWithPartialMaps()
elif R.category() is self.category():
cat = R.category()
else:
cat = EuclideanDomains() & MetricSpaces().Complete()
elif isinstance(R, Order) and R.number_field().defining_polynomial(
) == self.defining_polynomial():
cat = IntegralDomains()
elif isinstance(R, NumberField) and R.defining_polynomial(
) == self.defining_polynomial():
if self.is_field():
cat = Fields()
else:
cat = SetsWithPartialMaps()
else:
k = self.residue_field()
if R is k:
return ResidueLiftingMap._create_(R, self)
if cat is not None:
H = Hom(R, self, cat)
return H.__make_element_class__(DefPolyConversion)(H)