(f, u, K)
return s, f, r
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x, y = ring("x,y", K)
    >>> _, X, Y = ring("x,y", QQ)

    >>> s, f, r = R.dmp_sqf_norm(x*y + y**2)

    >>> s == 1
    True
    >>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
    True
    >>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
    True
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Square - free norm of f in K [ X ] useful over algebraic domains .
def dmp_sqf_norm(f, u, K):
    """
    Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x, y = ring("x,y", K)
    >>> _, X, Y = ring("x,y", QQ)

    >>> s, f, r = R.dmp_sqf_norm(x*y + y**2)

    >>> s == 1
    True
    >>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
    True
    >>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
    True

    """
    if not u:
        return dup_sqf_norm(f, K)

    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    F = dmp_raise([K.one, -K.unit], u, 0, K)

    s = 0

    while True:
        h, _ = dmp_inject(f, u, K, front=True)
        r = dmp_resultant(g, h, u + 1, K.dom)

        if dmp_sqf_p(r, u, K.dom):
            break
        else:
            f, s = dmp_compose(f, F, u, K), s + 1

    return s, f, r
def dmp_sqf_norm ( f , u , K ) : if not u : return dup_sqf_norm ( f , K ) if not K . is_Algebraic : raise DomainError ( "ground domain must be algebraic" ) g = dmp_raise ( K . mod . rep , u + 1 , 0 , K . dom ) F = dmp_raise ( [ K . one , - K . unit ] , u , 0 , K ) s = 0 while True : h , _ = dmp_inject ( f , u , K , front = True ) r = dmp_resultant ( g , h , u + 1 , K . dom ) if dmp_sqf_p ( r , u , K . dom ) : break else : f , s = dmp_compose ( f , F , u , K ) , s + 1 return s , f , r