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_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(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
