Univariate polynomials over number fields - Polynomials

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# Univariate polynomials over number fields

AUTHOR:

- Luis Felipe Tabera Alonso (2014-02): initial version.

EXAMPLES:

Define a polynomial over an absolute number field and perform basic operations with them:

Sage

    sage: x = polygen(ZZ, 'x')
    sage: N.<a> = NumberField(x^2 - 2)
    sage: K.<x> = N[]
    sage: f = x - a
    sage: g = x^3 - 2*a + 1
    sage: f * (x + a)
    x^2 - 2
    sage: f + g
    x^3 + x - 3*a + 1
    sage: g // f
    x^2 + a*x + 2
    sage: g % f
    1
    sage: factor(x^3 - 2*a*x^2 - 2*x + 4*a)
    (x - 2*a) * (x - a) * (x + a)
    sage: gcd(f, x - a)
    x - a

Python

    >>> from sage.all import *
    >>> x = polygen(ZZ, 'x')
    >>> N = NumberField(x**Integer(2) - Integer(2), names=('a',)); (a,) = N._first_ngens(1)
    >>> K = N['x']; (x,) = K._first_ngens(1)
    >>> f = x - a
    >>> g = x**Integer(3) - Integer(2)*a + Integer(1)
    >>> f * (x + a)
    x^2 - 2
    >>> f + g
    x^3 + x - 3*a + 1
    >>> g // f
    x^2 + a*x + 2
    >>> g % f
    1
    >>> factor(x**Integer(3) - Integer(2)*a*x**Integer(2) - Integer(2)*x + Integer(4)*a)
    (x - 2*a) * (x - a) * (x + a)
    >>> gcd(f, x - a)
    x - a

Polynomials are aware of embeddings of the underlying field:

Sage

    sage: x = polygen(ZZ, 'x')
    sage: Q7 = Qp(7)
    sage: r1 = Q7(3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8
    ....:          + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17
    ....:          + 4*7^18 + 6*7^19)
    sage: N.<b> = NumberField(x^2 - 2, embedding=r1)
    sage: K.<t> = N[]
    sage: f = t^3 - 2*t + 1
    sage: f(r1)
    1 + O(7^20)

[Python]

    >>> from sage.all import *
    >>> x = polygen(ZZ, 'x')
    >>> Q7 = Qp(Integer(7))
    >>> r1 = Q7(Integer(3) + Integer(7) + Integer(2)*Integer(7)**Integer(2) + Integer(6)*Integer(7)**Integer(3) + Integer(7)**Integer(4) + Integer(2)*Integer(7)**Integer(5) + Integer(7)**Integer(6) + Integer(2)*Integer(7)**Integer(7) + Integer(4)*Integer(7)**Integer(8)
    ...          + Integer(6)*Integer(7)**Integer(9) + Integer(6)*Integer(7)**Integer(10) + Integer(2)*Integer(7)**Integer(11) + Integer(7)**Integer(12) + Integer(7)**Integer(13) + Integer(2)*Integer(7)**Integer(15) + Integer(7)**Integer(16) + Integer(7)**Integer(17)
    ...          + Integer(4)*Integer(7)**Integer(18) + Integer(6)*Integer(7)**Integer(19))
    >>> N = NumberField(x**Integer(2) - Integer(2), embedding=r1, names=('b',)); (b,) = N._first_ngens(1)
    >>> K = N['t']; (t,) = K._first_ngens(1)
    >>> f = t**Integer(3) - Integer(2)*t + Integer(1)
    >>> f(r1)
    1 + O(7^20)

We can also construct polynomials over relative number fields:

Sage

    sage: N.<i, s2> = QQ[I, sqrt(2)]
    sage: K.<x> = N[]
    sage: f = x - s2
    sage: g = x^3 - 2*i*x^2 + s2*x
    sage: f * (x + s2)
    x^2 - 2
    sage: f + g
    x^3 - 2*I*x^2 + (sqrt2 + 1)*x - sqrt2
    sage: g // f
    x^2 + (-2*I + sqrt2)*x - 2*sqrt2*I + sqrt2 + 2
    sage: g % f
    -4*I + 2*sqrt2 + 2
    sage: factor(i*x^4 - 2*i*x^2 + 9*i)
    (I) * (x - I + sqrt2) * (x + I - sqrt2) * (x - I - sqrt2) * (x + I + sqrt2)
    sage: gcd(f, x - i)
    1

Python

    >>> from sage.all import *
    >>> N = QQ[I, sqrt(Integer(2))]; (i, s2,) = N._first_ngens(2)
    >>> K = N['x']; (x,) = K._first_ngens(1)
    >>> f = x - s2
    >>> g = x**Integer(3) - Integer(2)*i*x**Integer(2) + s2*x
    >>> f * (x + s2)
    x^2 - 2
    >>> f + g
    x^3 - 2*I*x^2 + (sqrt2 + 1)*x - sqrt2
    >>> g // f
    x^2 + (-2*I + sqrt2)*x - 2*sqrt2*I + sqrt2 + 2
    >>> g % f
    -4*I + 2*sqrt2 + 2
    >>> factor(i*x**Integer(4) - Integer(2)*i*x**Integer(2) + Integer(9)*i)
    (I) * (x - I + sqrt2) * (x + I - sqrt2) * (x - I - sqrt2) * (x + I + sqrt2)
    >>> gcd(f, x - i)
    1

class sage.rings.polynomial.polynomial_number_field.Polynomial_absolute_number_field_dense(parent, x=None, check=True, is_gen=False, construct=False)

  Bases: `Polynomial_generic_dense_field`

  Class of dense univariate polynomials over an absolute number field.

  gcd(other)

    Compute the monic gcd of two univariate polynomials using PARI.

    INPUT:

    - `other` – a polynomial with the same parent as `self`

    OUTPUT: the monic gcd of `self` and `other`

    EXAMPLES:

    Sage

        sage: x = polygen(ZZ, 'x')
        sage: N.<a> = NumberField(x^3 - 1/2, 'a')
        sage: R.<r> = N['r']
        sage: f = (5/4*a^2 - 2*a + 4)*r^2 + (5*a^2 - 81/5*a - 17/2)*r + 4/5*a^2 + 24*a + 6
        sage: g = (5/4*a^2 - 2*a + 4)*r^2 + (-11*a^2 + 79/5*a - 7/2)*r - 4/5*a^2 - 24*a - 6
        sage: gcd(f, g**2)
        r - 60808/96625*a^2 - 69936/96625*a - 149212/96625
        sage: R = QQ[I]['x']
        sage: f = R.random_element(2)
        sage: g = f + 1
        sage: h = R.random_element(2).monic()
        sage: f *= h
        sage: g *= h
        sage: gcd(f, g) - h
        0
        sage: f.gcd(g) - h
        0

    Python

        >>> from sage.all import *
        >>> x = polygen(ZZ, 'x')
        >>> N = NumberField(x**Integer(3) - Integer(1)/Integer(2), 'a', names=('a',)); (a,) = N._first_ngens(1)
        >>> R = N['r']; (r,) = R._first_ngens(1)
        >>> f = (Integer(5)/Integer(4)*a**Integer(2) - Integer(2)*a + Integer(4))*r**Integer(2) + (Integer(5)*a**Integer(2) - Integer(81)/Integer(5)*a - Integer(17)/Integer(2))*r + Integer(4)/Integer(5)*a**Integer(2) + Integer(24)*a + Integer(6)
        >>> g = (Integer(5)/Integer(4)*a**Integer(2) - Integer(2)*a + Integer(4))*r**Integer(2) + (-Integer(11)*a**Integer(2) + Integer(79)/Integer(5)*a - Integer(7)/Integer(2))*r - Integer(4)/Integer(5)*a**Integer(2) - Integer(24)*a - Integer(6)
        >>> gcd(f, g**Integer(2))
        r - 60808/96625*a^2 - 69936/96625*a - 149212/96625
        >>> R = QQ[I]['x']
        >>> f = R.random_element(Integer(2))
        >>> g = f + Integer(1)
        >>> h = R.random_element(Integer(2)).monic()
        >>> f *= h
        >>> g *= h
        >>> gcd(f, g) - h
        0
        >>> f.gcd(g) - h
        0

class sage.rings.polynomial.polynomial_number_field.Polynomial_relative_number_field_dense(parent, x=None, check=True, is_gen=False, construct=False)

  Bases: `Polynomial_generic_dense_field`

  Class of dense univariate polynomials over a relative number field.

  gcd(other)

    Compute the monic gcd of two polynomials.

    Currently, the method checks corner cases in which one of the polynomials is zero or a constant. Then, computes an absolute extension and performs the computations there.

    INPUT:

    - `other` – a polynomial with the same parent as `self`

    OUTPUT: the monic gcd of `self` and `other`

    See `Polynomial_absolute_number_field_dense.gcd()` for more details.

    EXAMPLES:

    Sage

        sage: N = QQ[sqrt(2), sqrt(3)]
        sage: s2, s3 = N.gens()
        sage: x = polygen(N)
        sage: f = x^4 - 5*x^2 + 6
        sage: g = x^3 + (-2*s2 + s3)*x^2 + (-2*s3*s2 + 2)*x + 2*s3
        sage: gcd(f, g)
        x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2
        sage: f.gcd(g)
        x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2

    Python

        >>> from sage.all import *
        >>> N = QQ[sqrt(Integer(2)), sqrt(Integer(3))]
        >>> s2, s3 = N.gens()
        >>> x = polygen(N)
        >>> f = x**Integer(4) - Integer(5)*x**Integer(2) + Integer(6)
        >>> g = x**Integer(3) + (-Integer(2)*s2 + s3)*x**Integer(2) + (-Integer(2)*s3*s2 + Integer(2))*x + Integer(2)*s3
        >>> gcd(f, g)
        x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2
        >>> f.gcd(g)
        x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2