Пример #1
0
        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
Пример #2
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        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
Пример #4
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    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
Пример #5
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    def super_categories(self):
        """
        EXAMPLES::

            sage: DiscreteValuationRings().super_categories()
            [Category of euclidean domains]
        """
        return [EuclideanDomains()]
Пример #6
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    def extra_super_categories(self):
        """
        EXAMPLES::

            sage: Fields().extra_super_categories()
            [Category of euclidean domains]

        """
        return [EuclideanDomains()]
Пример #7
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    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()]
Пример #8
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    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)