"""
Optimized Quadratic Number Field Elements

This file defines a Cython class ``NumberFieldElement_quadratic`` to speed up
computations in quadratic extensions of `\QQ`.

AUTHORS:

- Robert Bradshaw (2007-09): Initial version
- David Harvey (2007-10): fix up a few bugs, polish around the edges
- David Loeffler (2009-05): add more documentation and tests
- Vincent Delecroix (2012-07): comparisons for quadratic number fields
  (:trac:`13213`), abs, floor and ceil functions (:trac:`13256`)

.. TODO::

    The ``_new()`` method should be overridden in this class to copy the ``D``
    and ``standard_embedding`` attributes
"""

#*****************************************************************************
#       Copyright (C) 2007 Robert Bradshaw <robertwb@math.washington.edu>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from __future__ import absolute_import, division

include "sage/libs/ntl/decl.pxi"
from cpython.object cimport Py_EQ, Py_NE, Py_LE, Py_GE, Py_LT, Py_GT

from cysignals.signals cimport sig_on, sig_off

from sage.libs.gmp.mpz cimport *
from sage.libs.gmp.mpq cimport *
from sage.libs.ntl.ntl_ZZ cimport ntl_ZZ
from sage.libs.ntl.ntl_ZZX cimport ntl_ZZX
from sage.libs.mpfi cimport mpfi_set_z, mpfi_set_q, mpfi_sqrt, mpfi_add_z, mpfi_mul_z, mpfi_div_z, mpfi_neg

from sage.structure.parent_base cimport ParentWithBase
from sage.structure.element cimport Element
from sage.structure.richcmp cimport rich_to_bool_sgn

from sage.rings.rational cimport Rational
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_double import RDF
from sage.rings.complex_double import CDF
from sage.categories.morphism cimport Morphism
from sage.rings.number_field.number_field_element import _inverse_mod_generic
from sage.rings.real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class

from sage.libs.gmp.pylong cimport mpz_pythonhash


def __make_NumberFieldElement_quadratic0(parent, a, b, denom):
    """
    Used in unpickling elements of number fields.

    TESTS::

        sage: K.<a> = NumberField(x^2-x+13)
        sage: loads(dumps(a)) == a # indirect doctest
        True
    """
    return NumberFieldElement_quadratic(parent, (a, b, denom))

def __make_NumberFieldElement_quadratic1(parent, cls, a, b, denom):
    """
    Used in unpickling elements of number fields.

    TESTS::

        sage: K.<a> = NumberField(x^2-x+13)
        sage: loads(dumps(a)) == a # indirect doctest
        True

    We test that :trac:`6462` is fixed::

        sage: L = QuadraticField(-11,'a'); OL = L.maximal_order(); w = OL.0
        sage: loads(dumps(w)) == w # indirect doctest
        True
    """
    return cls(parent, (a, b, denom))


cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
    r"""
    A NumberFieldElement_quadratic object gives an efficient representation of
    an element of a quadratic extension of `\QQ`.

    Elements are represented internally as triples `(a, b, c)` of integers,
    where `{\rm gcd}(a, b, c) = 1` and `c > 0`, representing the element `(a +
    b \sqrt{D}) / c`. Note that if the discriminant `D` is `1 \bmod 4`,
    integral elements do not necessarily have `c = 1`.

    TESTS::

        sage: from sage.rings.number_field.number_field_element_quadratic import NumberFieldElement_quadratic

    We set up some fields::

        sage: K.<a> = NumberField(x^2+23)
        sage: a.parts()
        (0, 1)
        sage: F.<b> = NumberField(x^2-x+7)
        sage: b.parts()
        (1/2, 3/2)

    We construct elements of these fields in various ways - firstly, from
    polynomials::

        sage: NumberFieldElement_quadratic(K, x-1)
        a - 1
        sage: NumberFieldElement_quadratic(F, x-1)
        b - 1

    From triples of Integers::

        sage: NumberFieldElement_quadratic(K, (1,2,3))
        2/3*a + 1/3
        sage: NumberFieldElement_quadratic(F, (1,2,3))
        4/9*b + 1/9
        sage: NumberFieldElement_quadratic(F, (1,2,3)).parts()
        (1/3, 2/3)

    From pairs of Rationals::

        sage: NumberFieldElement_quadratic(K, (1/2,1/3))
        1/3*a + 1/2
        sage: NumberFieldElement_quadratic(F, (1/2,1/3))
        2/9*b + 7/18
        sage: NumberFieldElement_quadratic(F, (1/2,1/3)).parts()
        (1/2, 1/3)

    Direct from Rationals::

        sage: NumberFieldElement_quadratic(K, 2/3)
        2/3
        sage: NumberFieldElement_quadratic(F, 2/3)
        2/3

    This checks a bug when converting from lists::

        sage: w = CyclotomicField(3)([1/2,1])
        sage: w == w.__invert__().__invert__()
        True
    """

    def __init__(self, parent, f):
        """
        Standard initialisation function.

        EXAMPLES::

            sage: F.<a> = QuadraticField(-7)
            sage: c = a + 7
            sage: type(c) # indirect doctest
            <type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>
        """
        self.D = parent._D
        cdef Integer a, b, denom
        cdef Rational ad, bd

        cdef NumberFieldElement_quadratic gen

        if isinstance(f, NumberFieldElement_quadratic):
            self._parent = parent   # NOTE: We do *not* call NumberFieldElement_absolute.__init__, for speed reasons.
            mpz_set(self.a, (<NumberFieldElement_quadratic>f).a)
            mpz_set(self.b, (<NumberFieldElement_quadratic>f).b)
            mpz_set(self.denom, (<NumberFieldElement_quadratic>f).denom)

        elif type(f) is tuple and len(f) == 2:
            NumberFieldElement_absolute.__init__(self, parent, None)
            ad, bd = f
            mpz_lcm(self.denom, mpq_denref(ad.value), mpq_denref(bd.value))
            mpz_divexact(self.a, self.denom, mpq_denref(ad.value))
            mpz_mul(self.a, self.a, mpq_numref(ad.value))
            mpz_divexact(self.b, self.denom, mpq_denref(bd.value))
            mpz_mul(self.b, self.b, mpq_numref(bd.value))

        elif type(f) is tuple and len(f) == 3:
            NumberFieldElement_absolute.__init__(self, parent, None)
            a, b, denom = f
            mpz_set(self.a, a.value)
            mpz_set(self.b, b.value)
            mpz_set(self.denom, denom.value)
            self._reduce_c_()

        else:
            NumberFieldElement_absolute.__init__(self, parent, f)
            # poly is in gen (which may not be sqrt(d))
            self._ntl_coeff_as_mpz(self.a, 0)
            self._ntl_coeff_as_mpz(self.b, 1)
            if mpz_cmp_ui(self.a, 0) or mpz_cmp_ui(self.b, 0):
                gen = parent.gen()  # should this be cached?
                self._ntl_denom_as_mpz(self.denom)
                if mpz_cmp_ui(self.b, 0):
                    mpz_mul(self.a, self.a, gen.denom)
                    mpz_addmul(self.a, self.b, gen.a)
                    mpz_mul(self.b, self.b, gen.b)
                    mpz_mul(self.denom, self.denom, gen.denom)
            else:
                mpz_set_ui(self.denom, 1)
            self._reduce_c_()

        # set the attribute standard embedding which is used in the methods
        # __cmp__, sign, real, imag, floor, ceil, ...
        self.standard_embedding = parent._standard_embedding

    cdef _new(self):
        """
        Quickly creates a new initialized NumberFieldElement_quadratic with the
        same parent as self.

        EXAMPLES::

            sage: F.<b> = CyclotomicField(3)
            sage: b + b # indirect doctest
            2*b
        """
        cdef type t = type(self)
        cdef NumberFieldElement_quadratic x = <NumberFieldElement_quadratic>t.__new__(t)
        x._parent = self._parent
        x.standard_embedding = self.standard_embedding
        x.D = self.D
        return x

    cdef number_field(self):
        r"""
        Return the number field to which this element belongs. Since this is a
        Cython cdef method, it is not directly accessible by the user, but the
        function "_number_field" calls this one.

        EXAMPLES::

            sage: F.<b> = QuadraticField(-7)
            sage: b._number_field() # indirect doctest
            Number Field in b with defining polynomial x^2 + 7
        """
        return self._parent

    def _maxima_init_(self, I=None):
        """
        EXAMPLES::

            sage: K.<a> = QuadraticField(-1)
            sage: f = 1 + a
            sage: f._maxima_init_()
            '1+%i*1'
        """
        a = self.parent().gen()
        if a**2 == -1:
            x0, x1 = self
            return str(x0) + "+" + "%i*" + str(x1)
        else:
            NumberFieldElement_absolute._maxima_init_(self, I)

    def _polymake_init_(self):
        """
        EXAMPLES::

            sage: K.<sqrt5> = QuadraticField(5)
            sage: polymake(3+2*sqrt5)                 # optional - polymake
            3+2r5
            sage: K.<i> = QuadraticField(-1)
            sage: polymake(i)                         # optional - polymake
            Traceback (most recent call last):
            ...
            TypeError: Negative values for the root of the extension ... Bad Thing...
        """
        x0, x1 = self
        return "new QuadraticExtension({}, {}, {})".format(x0, x1, self.D)

    def __copy__(self):
        r"""
        Returns a new copy of self.

        TESTS::

            sage: K.<a> = QuadraticField(-3)
            sage: b = a + 3
            sage: c = b.__copy__()
            sage: b
            a + 3
            sage: c
            a + 3
            sage: b is c
            False
            sage: b == c
            True
        """
        cdef NumberFieldElement_quadratic x = <NumberFieldElement_quadratic>self._new()
        mpz_set(x.a, self.a)
        mpz_set(x.b, self.b)
        mpz_set(x.denom, self.denom)
        return x


    def __cinit__(self):
        r"""
        Initialisation function.

        EXAMPLES::

            sage: QuadraticField(-3, 'a').gen() # indirect doctest
            a
        """
        mpz_init(self.a)
        mpz_init(self.b)
        mpz_init(self.denom)


    def __dealloc__(self):
        mpz_clear(self.a)
        mpz_clear(self.b)
        mpz_clear(self.denom)


    def __reduce__(self):
        """
        Used for pickling.

        TESTS:

            sage: K.<a> = NumberField(x^2-13)
            sage: loads(dumps(a)) == a
            True
            sage: loads(dumps(a/3+5)) == a/3+5
            True
        """
        cdef Integer a = Integer.__new__(Integer)
        cdef Integer b = Integer.__new__(Integer)
        cdef Integer denom = Integer.__new__(Integer)
        mpz_set(a.value, self.a)
        mpz_set(b.value, self.b)
        mpz_set(denom.value, self.denom)
        return __make_NumberFieldElement_quadratic1, (self._parent, type(self), a, b, denom)

    cdef int _randomize(self, num_bound, den_bound, distribution) except -1:
        cdef Integer temp, denom1, denom2

        # in theory, we could just generate two random numerators and
        # a random denominator. however, this would mean that we were
        # extraordinarily unlikely to run into results of the form
        # 1/3 + 1/5*sqrt(D), which are often some of the best examples
        # for testing out code. since this is probably the primary use
        # of the random element code, it's worth doing slightly more
        # work to make this possible.

        # normalize denominator bound
        if den_bound is None or den_bound < 1:
            den_bound = 1

        # generate denominators
        denom1 = <Integer>(ZZ.random_element(x=1,
                                             y=den_bound+1,
                                             distribution=distribution))
        denom2 = <Integer>(ZZ.random_element(x=1,
                                             y=den_bound+1,
                                             distribution=distribution))

        # set a, b
        temp = <Integer>(ZZ.random_element(x=num_bound, distribution=distribution))
        mpz_mul(self.a, temp.value, denom2.value)
        temp = <Integer>(ZZ.random_element(x=num_bound, distribution=distribution))
        mpz_mul(self.b, temp.value, denom1.value)
        # set denom
        mpz_mul(self.denom, denom1.value, denom2.value)

        self._reduce_c_()
        return 0  # No error


    def _lift_cyclotomic_element(self, new_parent, bint check=True, int rel=0):
        """
        Creates an element of the passed field from this field.  This
        is specific to creating elements in a cyclotomic field from
        elements in another cyclotomic field, in the case that
        self.number_field()._n() divides new_parent()._n().  This
        function aims to make this common coercion extremely fast!

        More general coercion (i.e. of zeta6 into CyclotomicField(3))
        is implemented in the _coerce_from_other_cyclotomic_field
        method of a CyclotomicField.

        EXAMPLES::

            sage: C.<zeta4>=CyclotomicField(4)
            sage: CyclotomicField(20)(zeta4+1)  # The function _lift_cyclotomic_element does the heavy lifting in the background
            zeta20^5 + 1
            sage: (zeta4+1)._lift_cyclotomic_element(CyclotomicField(40))  # There is rarely a purpose to call this function directly
            zeta40^10 + 1

            sage: cf3 = CyclotomicField(3) ; z3 = cf3.0
            sage: cf6 = CyclotomicField(6) ; z6 = cf6.0
            sage: z6._lift_cyclotomic_element(cf3)
            Traceback (most recent call last):
            ...
            TypeError: The zeta_order of the new field must be a multiple of the zeta_order of the original.
            sage: cf3(z6)
            zeta3 + 1
            sage: z3._lift_cyclotomic_element(cf6)
            zeta6 - 1

        Verify embeddings are respected::

            sage: cf6c = CyclotomicField(6, embedding=CDF(exp(-pi*I/3))) ; z6c = cf6c.0
            sage: cf3(z6c)
            -zeta3
            sage: cf6c(z3)
            -zeta6

        AUTHOR:

        - Joel B. Mohler (original version)

        - Craig Citro (reworked for quadratic elements)
        """
        if check:
            from .number_field import NumberField_cyclotomic
            if not isinstance(self.number_field(), NumberField_cyclotomic) \
                   or not isinstance(new_parent, NumberField_cyclotomic):
                raise TypeError("The field and the new parent field must both be cyclotomic fields.")

        if rel == 0:
            small_order = self.number_field()._n()
            large_order = new_parent._n()

            try:
                rel = ZZ(large_order / small_order)
            except TypeError:
                raise TypeError("The zeta_order of the new field must be a multiple of the zeta_order of the original.")

        cdef NumberFieldElement_quadratic x2
        cdef int n = self._parent._n()

        if new_parent.degree() == 2:
            ## since self is a *quadratic* element, we can only get
            ## here if self.parent() and new_parent are:
            ## - CyclotomicField(3) and CyclotomicField(6)
            ## - CyclotomicField(3) and CyclotomicField(3)
            ## - CyclotomicField(6) and CyclotomicField(6)
            ## - CyclotomicField(4) and CyclotomicField(4)
            ## In all cases, conversion of elements is trivial!
            if n == <int>new_parent._n():
                conjugate = rel != 1
            else:
                # n = 3, new_n = 6
                conjugate = rel == 4
            x2 = <NumberFieldElement_quadratic>(self._new())
            x2._parent = new_parent
            mpz_set(x2.a, self.a)
            if conjugate:
                mpz_neg(x2.b, self.b)
            else:
                mpz_set(x2.b, self.b)
            mpz_set(x2.denom, self.denom)
            x2.D = self.D
            return x2

        cdef NumberFieldElement x
        cdef ZZX_c elt_num
        cdef ZZ_c elt_den, tmp_coeff
        cdef mpz_t tmp_mpz
        cdef long tmp_const

        x = <NumberFieldElement_absolute>NumberFieldElement_absolute.__new__(NumberFieldElement_absolute)

        mpz_to_ZZ(&elt_den, self.denom)

        mpz_init(tmp_mpz)

        ## set the two terms in the polynomial
        if n == 4:
            mpz_to_ZZ(&tmp_coeff, self.a)
            ZZX_SetCoeff(elt_num, 0, tmp_coeff)
            mpz_to_ZZ(&tmp_coeff, self.b)
            ZZX_SetCoeff(elt_num, 1, tmp_coeff)

        elif n == 3:
            ## num[0] = a + b
            mpz_add(tmp_mpz, tmp_mpz, self.a)
            mpz_add(tmp_mpz, tmp_mpz, self.b)
            mpz_to_ZZ(&tmp_coeff, tmp_mpz)
            ZZX_SetCoeff(elt_num, 0, tmp_coeff)

            ## num[1] = 2*b
            mpz_sub(tmp_mpz, tmp_mpz, self.a)
            tmp_const = 2
            mpz_mul_si(tmp_mpz, tmp_mpz, tmp_const)
            mpz_to_ZZ(&tmp_coeff, tmp_mpz)
            ZZX_SetCoeff(elt_num, 1, tmp_coeff)

        elif n == 6:
            ## num[0] = a - b
            mpz_add(tmp_mpz, tmp_mpz, self.a)
            mpz_sub(tmp_mpz, tmp_mpz, self.b)
            mpz_to_ZZ(&tmp_coeff, tmp_mpz)
            ZZX_SetCoeff(elt_num, 0, tmp_coeff)

            ## num[1] = 2*b
            mpz_sub(tmp_mpz, tmp_mpz, self.a)
            tmp_const = -2
            mpz_mul_si(tmp_mpz, tmp_mpz, tmp_const)
            mpz_to_ZZ(&tmp_coeff, tmp_mpz)
            ZZX_SetCoeff(elt_num, 1, tmp_coeff)

        mpz_clear(tmp_mpz)

        x._parent = <ParentWithBase>new_parent
        x.__fld_numerator, x.__fld_denominator = new_parent.polynomial_ntl()
        x.__denominator = elt_den
        cdef ZZX_c result
        cdef ZZ_c tmp
        cdef int i
        cdef ntl_ZZX _num
        cdef ntl_ZZ _den
        _num, _den = new_parent.polynomial_ntl()
        for i from 0 <= i <= ZZX_deg(elt_num):
            tmp = ZZX_coeff(elt_num, i)
            ZZX_SetCoeff(result, i*rel, tmp)
        ZZX_rem(x.__numerator, result, _num.x)
        (<NumberFieldElement_absolute>x)._reduce_c_()
        return x

    def _real_mpfi_(self, RealIntervalField_class R):
        r"""
        Conversion to a real interval field

        TESTS::

            sage: K1 = QuadraticField(2)
            sage: RIF(K1.gen())
            1.414213562373095?
            sage: RIF(K1(1/5))
            0.2000000000000000?
            sage: RIF(3/5*K1.gen() + 1/5)
            1.048528137423857?

            sage: K2 = QuadraticField(2, embedding=-RLF(2).sqrt())
            sage: RIF(K2.gen())
            -1.414213562373095?

            sage: K3 = QuadraticField(-1)
            sage: RIF(K3.gen())
            Traceback (most recent call last):
            ...
            ValueError: can not convert complex algebraic number to real interval
            sage: RIF(K3(2))
            2

        Check that :trac:`21979` is fixed::

            sage: a = QuadraticField(5).gen()
            sage: u = -573147844013817084101/2*a + 1281597540372340914251/2
            sage: RealIntervalField(128)(u)
            0.?e-17
            sage: RealIntervalField(128)(u).is_zero()
            False
        """
        cdef RealIntervalFieldElement ans = R._new()

        if mpz_cmp_ui(self.b, 0):
            if mpz_cmp_ui(self.D.value, 0) < 0:
                raise ValueError("can not convert complex algebraic number to real interval")
            mpfi_set_z(ans.value, self.D.value)
            mpfi_sqrt(ans.value, ans.value)
            if not self.standard_embedding:
                mpfi_neg(ans.value, ans.value)
            mpfi_mul_z(ans.value, ans.value, self.b)

            mpfi_add_z(ans.value, ans.value, self.a)

        else:
            mpfi_set_z(ans.value, self.a)

        mpfi_div_z(ans.value, ans.value, self.denom)
        return ans

    def parts(self):
        """
        This function returns a pair of rationals `a` and `b` such that self `=
        a+b\sqrt{D}`.

        This is much closer to the internal storage format of the
        elements than the polynomial representation coefficients (the output of
        ``self.list()``), unless the generator with which this number field was
        constructed was equal to `\sqrt{D}`. See the last example below.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-13)
            sage: K.discriminant()
            13
            sage: a.parts()
            (0, 1)
            sage: (a/2-4).parts()
            (-4, 1/2)
            sage: K.<a> = NumberField(x^2-7)
            sage: K.discriminant()
            28
            sage: a.parts()
            (0, 1)
            sage: K.<a> = NumberField(x^2-x+7)
            sage: a.parts()
            (1/2, 3/2)
            sage: a._coefficients()
            [0, 1]
        """
        cdef Rational ad = <Rational>Rational.__new__(Rational)
        cdef Rational bd = <Rational>Rational.__new__(Rational)
        if mpz_cmp_ui(self.a, 0) == 0:
            mpq_set_ui(ad.value, 0, 1)
        else:
            mpz_set(mpq_numref(ad.value), self.a)
            mpz_set(mpq_denref(ad.value), self.denom)
            mpq_canonicalize(ad.value)
        if mpz_cmp_ui(self.b, 0) == 0:
            mpq_set_ui(bd.value, 0, 1)
        else:
            mpz_set(mpq_numref(bd.value), self.b)
            mpz_set(mpq_denref(bd.value), self.denom)
            mpq_canonicalize(bd.value)

        return ad, bd

    cdef bint is_sqrt_disc(self):
        r"""
        Returns true if self is `\sqrt{D}`.

        EXAMPLES::

            sage: F.<b> = NumberField(x^2 - x + 7)
            sage: b.denominator() # indirect doctest
            1
        """
        return mpz_cmp_ui(self.denom, 1)==0 and mpz_cmp_ui(self.a, 0)==0 and mpz_cmp_ui(self.b, 1)==0

#########################################################
# Comparisons
#########################################################

    def sign(self):
        r"""
        Returns the sign of self (0 if zero, +1 if positive and -1 if negative).

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2, name='sqrt2')
            sage: K(0).sign()
            0
            sage: sqrt2.sign()
            1
            sage: (sqrt2+1).sign()
            1
            sage: (sqrt2-1).sign()
            1
            sage: (sqrt2-2).sign()
            -1
            sage: (-sqrt2).sign()
            -1
            sage: (-sqrt2+1).sign()
            -1
            sage: (-sqrt2+2).sign()
            1

            sage: K.<a> = QuadraticField(2, embedding=-1.4142)
            sage: K(0).sign()
            0
            sage: a.sign()
            -1
            sage: (a+1).sign()
            -1
            sage: (a+2).sign()
            1
            sage: (a-1).sign()
            -1
            sage: (-a).sign()
            1
            sage: (-a-1).sign()
            1
            sage: (-a-2).sign()
            -1

            sage: K.<b> = NumberField(x^2 + 2*x + 7, 'b', embedding=CC(-1,-sqrt(6)))
            sage: b.sign()
            Traceback (most recent call last):
            ...
            ValueError: a complex number has no sign!
            sage: K(1).sign()
            1
            sage: K(0).sign()
            0
            sage: K(-2/3).sign()
            -1
        """
        cdef mpz_t i, j
        cdef int s = 1, test

        if mpz_sgn(self.b) == 0:
            return mpz_sgn(self.a)

        if mpz_sgn(self.D.value) == -1:
            raise ValueError("a complex number has no sign!")

        if not self.standard_embedding:
            s = -1

        if mpz_sgn(self.a) == 0:
            return s*mpz_sgn(self.b)

        if mpz_sgn(self.a) == 1:
            if mpz_sgn(self.b) == s:
                return 1

        elif mpz_sgn(self.b) == -s:
            return -1

        mpz_init_set(i,self.a)
        mpz_mul(i,i,i)
        mpz_init_set(j,self.b)
        mpz_mul(j,j,j)
        mpz_mul(j,j,self.D.value)
        test = mpz_cmp(i,j)
        mpz_clear(i)
        mpz_clear(j)
        if mpz_sgn(self.a) == 1 and mpz_sgn(self.b) == -s:
            return test
        return -test

    cpdef _richcmp_(left, _right, int op):
        r"""
        Rich comparison of elements.

        TESTS::

            sage: K.<i> = QuadraticField(-1)
            sage: sorted([5*i+1, 2, 3*i+1, 2-i])
            [3*i + 1, 5*i + 1, -i + 2, 2]

        Make some random tests to check that the order is compatible with the
        ones of the real field (RR) and complex field (CC)::

            sage: K1 = NumberField(x^2 - 2, 'a', embedding=RR(1.4))
            sage: K2 = NumberField(x^2 - 2, 'a', embedding=RR(-1.4))
            sage: for _ in range(500):  # long time
            ....:     for K in K1, K2:
            ....:         a = K.random_element()
            ....:         b = K.random_element()
            ....:         assert (a < b) == (RR(a) < RR(b))
            ....:         assert (a > b) == (RR(a) > RR(b))
            ....:         assert (a == b) == (RR(a) == RR(b))
            ....:         assert (a != b) == (RR(a) != RR(b))
            ....:         assert (a >= b) == (RR(a) >= RR(b))
            ....:         assert (a <= b) == (RR(a) <= RR(b))

        ::

            sage: K1 = NumberField(x^2 + 2, 'a', embedding=CC(0,1))
            sage: K2 = NumberField(x^2 + 2, 'a', embedding=CC(0,-1))
            sage: for _ in range(500):  # long time
            ....:     for K in K1, K2:
            ....:         a = K.random_element()
            ....:         b = K.random_element()
            ....:         assert (a < b) == (CC(a) < CC(b))
            ....:         assert (a > b) == (CC(a) > CC(b))
            ....:         assert (a == b) == (CC(a) == CC(b))
            ....:         assert (a != b) == (CC(a) != CC(b))
            ....:         assert (a >= b) == (CC(a) >= CC(b))
            ....:         assert (a <= b) == (CC(a) <= CC(b))

        The following is tested because of the implementation of
        :func:`Q_to_quadratic_field_element` which was the cause of
        some problems with :trac:`13213`::

            sage: K.<sqrt2> = QuadraticField(2)
            sage: 1/2 + sqrt2 > 0
            True

        Two examples from the same number field with its two possible real
        embeddings::

            sage: K.<phi> = NumberField(x^2-x-1, 'phi', embedding=1.618)
            sage: phi > 0
            True
            sage: -phi > 0
            False
            sage: phi - 3 == 2*phi + 1
            False
            sage: fibonacci(10)*phi < fibonacci(11)
            True
            sage: RDF(fibonacci(10)*phi)
            88.99186938124421
            sage: fibonacci(11)
            89
            sage: l = [-2, phi+3, 2*phi-1, 2*phi-5, 0, -phi+2, fibonacci(20)*phi - fibonacci(21)]
            sage: l.sort()
            sage: l
            [-2, 2*phi - 5, 6765*phi - 10946, 0, -phi + 2, 2*phi - 1, phi + 3]
            sage: list(map(RDF, l))
            [-2.0, -1.7639320225002102, -6.610696073039435e-05, 0.0, 0.3819660112501051, 2.23606797749979, 4.618033988749895]

        ::

            sage: L.<psi> = NumberField(x^2-x-1, 'psi', embedding=-0.618)
            sage: psi < 0
            True
            sage: 2*psi + 3 == 2*psi + 3
            True
            sage: fibonacci(10)*psi < -fibonacci(9)
            False
            sage: RDF(fibonacci(10)*psi)
            -33.99186938124422
            sage: fibonacci(9)
            34
            sage: l = [-1, psi, 0, fibonacci(20)*psi + fibonacci(19), 3*psi+2]
            sage: l.sort()
            sage: l
            [-1, psi, 0, 6765*psi + 4181, 3*psi + 2]
            sage: list(map(RDF, l))
            [-1.0, -0.6180339887498949, 0.0, 6.610696073039435e-05, 0.1458980337503153]

        For a field with no specified embedding the comparison uses the standard
        embedding::

            sage: K.<sqrt2> = NumberField(x^2-2, 'sqrt2')
            sage: sqrt2 > 1 and sqrt2 < 2
            True

        The following examples illustrate the same behavior for a complex
        quadratic field::

            sage: K.<i> = QuadraticField(-1)
            sage: l = [-2, i-3, 2*i-2, 2*i+2, 5*i, 1-3*i, -1+i, 1]
            sage: l.sort()
            sage: l
            [i - 3, -2, 2*i - 2, i - 1, 5*i, -3*i + 1, 1, 2*i + 2]
            sage: list(map(CDF, l))
            [-3.0 + 1.0*I, -2.0, -2.0 + 2.0*I, -1.0 + 1.0*I, 5.0*I, 1.0 - 3.0*I, 1.0, 2.0 + 2.0*I]
            sage: list(map(CDF, l)) == sorted(map(CDF, l))
            True
        """
        # When D > 0 and standard embedding, we compare (a + b * sqrt(D)) / d and (aa +
        # bb * sqrt(D)) / dd using the comparison of (dd*a - d * aa)^2 and (d*bb - dd*b)^2 * D
        # mpz_sgn: returns 1 if > 0, 0 if 0 and -1 if < 0
        cdef mpz_t i, j
        cdef NumberFieldElement_quadratic right = <NumberFieldElement_quadratic> _right
        cdef int test

        # inequality and equality
        if mpz_cmp(left.a, right.a) or mpz_cmp(left.b, right.b) or mpz_cmp(left.denom, right.denom):
            if op == Py_EQ:
                return False
            elif op == Py_NE:
                return True
        else: # equality
            if op == Py_EQ or op == Py_LE or op == Py_GE:
                return True
            if op == Py_NE or op == Py_LT or op == Py_GT:
                return False

        # comparisons are valid only in *real* quadratic number field
        # when no embedding is specified or in the case of complex embeddings we
        # use a lexicographic order.
        if mpz_sgn(left.D.value) == -1:
            mpz_init(i)
            mpz_init(j)
            mpz_mul(i, left.a, right.denom)
            mpz_mul(j, right.a, left.denom)
            test = mpz_cmp(i,j)
            if test:
                mpz_clear(i)
                mpz_clear(j)
                return rich_to_bool_sgn(op, test)
            mpz_mul(i, left.b, right.denom)
            mpz_mul(j, right.b, left.denom)
            test = mpz_cmp(i,j)
            if test:
                if not left.standard_embedding:
                    test = -test
                mpz_clear(i)
                mpz_clear(j)
                return rich_to_bool_sgn(op, test)
            test = mpz_cmp(left.denom, right.denom)
            mpz_clear(i)
            mpz_clear(j)
            return rich_to_bool_sgn(op, test)

        # comparison in the real case
        mpz_init(i)
        mpz_mul(i,  right.denom, left.a)
        mpz_submul(i, left.denom, right.a)

        mpz_init(j)
        mpz_mul(j, left.denom, right.b)
        mpz_submul(j, right.denom, left.b)

        if not left.standard_embedding:
            mpz_neg(j, j)

        if mpz_sgn(i) == 1:
            if mpz_sgn(j) == 1:
                mpz_mul(i, i, i)
                mpz_mul(j, j, j)
                mpz_mul(j, j, left.D.value)
                test = mpz_cmp(i, j)
            else:
                test = 1

        else:
            if mpz_sgn(j) == -1:
                mpz_mul(i, i, i)
                mpz_mul(j, j, j)
                mpz_mul(j, j, left.D.value)
                test = mpz_cmp(j, i)
            else:
                test = -1

        mpz_clear(i)
        mpz_clear(j)
        return rich_to_bool_sgn(op, test)

    def continued_fraction_list(self):
        r"""
        Return the preperiod and the period of the continued fraction expansion
        of ``self``.

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2)
            sage: sqrt2.continued_fraction_list()
            ((1,), (2,))
            sage: (1/2+sqrt2/3).continued_fraction_list()
            ((0, 1, 33), (1, 32))

        For rational entries a pair of tuples is also returned but the second
        one is empty::

            sage: K(123/567).continued_fraction_list()
            ((0, 4, 1, 1, 1, 1, 3, 2), ())
        """
        cdef NumberFieldElement_quadratic x

        if mpz_sgn(self.b) == 0:
            return tuple(Rational(self).continued_fraction_list()),()

        if mpz_sgn(self.D.value) < 0:
            raise ValueError("the method is only available for positive discriminant")

        x = self
        orbit = []
        quots = []
        while x not in orbit:
            quots.append(x.floor())
            orbit.append(x)
            x = ~(x - quots[-1])

        i = orbit.index(x)

        return tuple(quots[:i]), tuple(quots[i:])

    def continued_fraction(self):
        r"""
        Return the (finite or ultimately periodic) continued fraction of ``self``.

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2)
            sage: cf = sqrt2.continued_fraction(); cf
            [1; (2)*]
            sage: cf.n()
            1.41421356237310
            sage: sqrt2.n()
            1.41421356237309
            sage: cf.value()
            sqrt2

            sage: (sqrt2/3 + 1/4).continued_fraction()
            [0; 1, (2, 1, 1, 2, 3, 2, 1, 1, 2, 5, 1, 1, 14, 1, 1, 5)*]
        """
        t1,t2 = self.continued_fraction_list()
        from sage.rings.continued_fraction import ContinuedFraction_periodic
        return ContinuedFraction_periodic(t1,t2)

#########################################################
# Arithmetic
#########################################################

    cdef void _reduce_c_(self):
        r"""
        Reduces into canonical form.

        WARNING: this mutates self.
        """
        cdef mpz_t gcd
        # cancel out common factors
        mpz_init(gcd)
        mpz_gcd(gcd, self.a, self.denom)
        mpz_gcd(gcd, gcd, self.b)
        if mpz_cmp_si(gcd, 1): # != 0 (i.e. it is not 1)
            mpz_divexact(self.a, self.a, gcd)
            mpz_divexact(self.b, self.b, gcd)
            mpz_divexact(self.denom, self.denom, gcd)
        # make denominator positive
        if mpz_sgn(self.denom) < 0:
            mpz_neg(self.denom, self.denom)
            mpz_neg(self.a, self.a)
            mpz_neg(self.b, self.b)
        mpz_clear(gcd)


    cpdef _add_(self, other_m):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2-5)
            sage: K.discriminant()
            5
            sage: a+a # indirect doctest
            2*a
            sage: s = (a+2)/6; s
            1/6*a + 1/3
            sage: s+a
            7/6*a + 1/3
            sage: s+10
            1/6*a + 31/3
            sage: s+(2*a+5)/7
            19/42*a + 22/21
            sage: s+(1+a)/2
            2/3*a + 5/6
            sage: s+(1+a)/8
            7/24*a + 11/24
            sage: s+(a+5)/6
            1/3*a + 7/6
            sage: (a/3+2/3) + (2*a/3+1/3)
            a + 1
        """
        cdef NumberFieldElement_quadratic other = <NumberFieldElement_quadratic>other_m
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        cdef mpz_t gcd, tmp
        if mpz_cmp(self.denom, other.denom) == 0:
            mpz_add(res.a, self.a, other.a)
            mpz_add(res.b, self.b, other.b)
            mpz_set(res.denom, self.denom)
        else:
            mpz_init(gcd)
            mpz_gcd(gcd, self.denom, other.denom)
            if mpz_cmp_ui(gcd, 1) == 0:
                mpz_mul(res.a, self.a, other.denom)
                mpz_addmul(res.a, self.denom, other.a)
                mpz_mul(res.b, self.b, other.denom)
                mpz_addmul(res.b, self.denom, other.b)
                mpz_mul(res.denom, self.denom, other.denom)
            else:
                mpz_init(tmp)
                mpz_divexact(tmp, other.denom, gcd)
                mpz_mul(res.a, self.a, tmp)
                mpz_mul(res.b, self.b, tmp)
                mpz_divexact(tmp, self.denom, gcd)
                mpz_addmul(res.a, other.a, tmp)
                mpz_addmul(res.b, other.b, tmp)
                mpz_mul(res.denom, other.denom, tmp)
                mpz_clear(tmp)
            mpz_clear(gcd)
        res._reduce_c_()
        return res


    cpdef _sub_(self, other_m):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2-13)
            sage: b = (a-3)/10; b # indirect doctest
            1/10*a - 3/10
            sage: b-1
            1/10*a - 13/10
            sage: b-a
            -9/10*a - 3/10
            sage: b-1/2
            1/10*a - 4/5
            sage: b-a/15
            1/30*a - 3/10
        """
        cdef NumberFieldElement_quadratic other = <NumberFieldElement_quadratic>other_m
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        cdef mpz_t gcd, tmp
        if mpz_cmp(self.denom, other.denom) == 0:
            mpz_sub(res.a, self.a, other.a)
            mpz_sub(res.b, self.b, other.b)
            mpz_set(res.denom, self.denom)
        else:
            mpz_init(gcd)
            mpz_gcd(gcd, self.denom, other.denom)
            if mpz_cmp_ui(gcd, 1) == 0:
                mpz_mul(res.a, self.a, other.denom)
                mpz_submul(res.a, self.denom, other.a)
                mpz_mul(res.b, self.b, other.denom)
                mpz_submul(res.b, self.denom, other.b)
                mpz_mul(res.denom, self.denom, other.denom)
            else:
                mpz_init(tmp)
                mpz_divexact(tmp, other.denom, gcd)
                mpz_mul(res.a, self.a, tmp)
                mpz_mul(res.b, self.b, tmp)
                mpz_divexact(tmp, self.denom, gcd)
                mpz_submul(res.a, other.a, tmp)
                mpz_submul(res.b, other.b, tmp)
                mpz_mul(res.denom, other.denom, tmp)
                mpz_clear(tmp)
            mpz_clear(gcd)
        res._reduce_c_()
        return res


    def __neg__(self):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+163)
            sage: -a
            -a
            sage: -(a+4)
            -a - 4
            sage: b = (a-3)/2
            sage: -b
            -1/2*a + 3/2
        """
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_neg(res.a, self.a)
        mpz_neg(res.b, self.b)
        mpz_set(res.denom, self.denom)
        return res

    cpdef _mul_(self, other_m):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+23)
            sage: a*a # indirect doctest
            -23
            sage: (a+1)*(a-1)
            -24
            sage: (a+1)*(a+2)
            3*a - 21
            sage: (a+1)/2 * (a+2)
            3/2*a - 21/2
            sage: (a+1)/2 * (a+2)/3
            1/2*a - 7/2
            sage: (2*a+4) * (3*a)/2
            6*a - 69

        Verify Karatsuba::

            sage: K.<a> = NumberField(x^2-41)
            sage: (10^1000 * (a+1)) * K(2+3*a) == 10^1000 * ((a+1) * K(2+3*a))
            True
        """
        cdef NumberFieldElement_quadratic other = <NumberFieldElement_quadratic>other_m
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        cdef mpz_t tmp

        if mpz_size(self.a) + mpz_size(self.b) < 8: # could I use a macro instead?
            # Do it the traditional way
            mpz_mul(res.a, self.b, other.b)
            mpz_mul(res.a, res.a, self.D.value)
            mpz_addmul(res.a, self.a, other.a)

            mpz_mul(res.b, self.a, other.b)
            mpz_addmul(res.b, self.b, other.a)

        else:
            # Karatsuba
            sig_on()
            mpz_init(tmp)
            mpz_add(res.a, self.a, self.b) # using res.a as tmp
            mpz_add(tmp, other.a, other.b)
            mpz_mul(res.b, res.a, tmp) # res.b = (self.a + self.b)(other.a + other.b)

            mpz_mul(res.a, self.a, other.a)
            mpz_sub(res.b, res.b, res.a)
            mpz_mul(tmp, self.b, other.b)
            mpz_sub(res.b, res.b, tmp)
            mpz_mul(tmp, tmp, self.D.value)
            mpz_add(res.a, res.a, tmp)
            mpz_clear(tmp)
            sig_off()

        mpz_mul(res.denom, self.denom, other.denom)
        res._reduce_c_()
        return res

    cpdef _rmul_(self, Element _c):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+43)
            sage: (1+a)*3 # indirect doctest
            3*a + 3
        """
        cdef Rational c =  <Rational>_c
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_mul(res.a, self.a, mpq_numref(c.value))
        mpz_mul(res.b, self.b, mpq_numref(c.value))
        mpz_mul(res.denom, self.denom, mpq_denref(c.value))
        res._reduce_c_()
        return res

    cpdef _lmul_(self, Element _c):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+43)
            sage: 5*(a-1/5) # indirect doctest
            5*a - 1
        """
        cdef Rational c =  <Rational>_c
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_mul(res.a, self.a, mpq_numref(c.value))
        mpz_mul(res.b, self.b, mpq_numref(c.value))
        mpz_mul(res.denom, self.denom, mpq_denref(c.value))
        res._reduce_c_()
        return res

    def __invert__(self):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2-5)
            sage: ~a
            1/5*a
            sage: ~(a+1)
            1/4*a - 1/4
            sage: (a-1)*(a+1)
            4
            sage: b = ~(5*a-3); b
            5/116*a + 3/116
            sage: b*(5*a-3)
            1
            sage: b = ~((3*a-2)/7); b
            21/41*a + 14/41
            sage: (3*a-2)/7 * b
            1

        This fixes ticket :trac:`9357`::

            sage: K.<a> = NumberField(x^2+1)
            sage: d = K(0)
            sage: ~d
            Traceback (most recent call last):
            ...
            ZeroDivisionError: number field element division by zero
            sage: K.random_element() / d
            Traceback (most recent call last):
            ...
            ZeroDivisionError: number field element division by zero
        """
        if mpz_cmp_ui(self.a, 0) == 0 and mpz_cmp_ui(self.b, 0) == 0:
            raise ZeroDivisionError("number field element division by zero")
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        cdef mpz_t tmp, gcd
        mpz_init(tmp)
        mpz_init(gcd)

        mpz_gcd(gcd, self.a, self.b)
        if mpz_cmp_si(gcd, 1): # != 0 (i.e. it is not 1)
            # cancel out g (g(a'-b'd)) / (g^2 (a'^2-b'^2d^2))
            mpz_divexact(res.a, self.a, gcd)
            mpz_divexact(res.b, self.b, gcd)
            mpz_neg(res.b, res.b)
        else:
            mpz_set(res.a, self.a)
            mpz_neg(res.b, self.b)

        mpz_pow_ui(res.denom, res.a, 2)
        mpz_pow_ui(tmp, res.b, 2)
        mpz_mul(tmp, tmp, self.D.value)
        mpz_sub(res.denom, res.denom, tmp)
        # need to multiply the leftover g back in
        mpz_mul(res.denom, res.denom, gcd)

        mpz_mul(res.a, res.a, self.denom)
        mpz_mul(res.b, res.b, self.denom)

        mpz_clear(tmp)
        mpz_clear(gcd)

        res._reduce_c_()
        return res

    cpdef NumberFieldElement galois_conjugate(self):
        """
        Return the image of this element under action of the nontrivial
        element of the Galois group of this field.

        EXAMPLES::

            sage: K.<a> = QuadraticField(23)
            sage: a.galois_conjugate()
            -a

            sage: K.<a> = NumberField(x^2 - 5*x + 1)
            sage: a.galois_conjugate()
            -a + 5
            sage: b = 5*a + 1/3
            sage: b.galois_conjugate()
            -5*a + 76/3
            sage: b.norm() ==  b * b.galois_conjugate()
            True
            sage: b.trace() ==  b + b.galois_conjugate()
            True
        """
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_set(res.a, self.a)
        mpz_neg(res.b, self.b)
        mpz_set(res.denom, self.denom)
        return res

#################################################################################
# We must override everything that makes uses of self.__numerator/__denominator
#################################################################################

    def __hash__(self):
        """
        Return hash of this number field element.

        For elements in `\ZZ` or `\QQ` the hash coincides with the one in the
        native `\ZZ` or `\QQ`.

        EXAMPLES::

            sage: L.<a> = QuadraticField(-7)
            sage: hash(a)
            42082631
            sage: hash(L(1))
            1
            sage: hash(L(-3))
            -3
            sage: hash(L(-32/118)) == hash(-32/118)
            True
        """
        # 1. compute the hash of a/denom as if it was a rational
        # (see the corresponding code in sage/rings/rational.pyx)
        cdef Py_hash_t n = mpz_pythonhash(self.a)
        cdef Py_hash_t d = mpz_pythonhash(self.denom)
        cdef Py_hash_t h = n + (d - 1) * <Py_hash_t>(7461864723258187525)

        # 2. mix the hash together with b
        h += 42082631 * mpz_pythonhash(self.b)
        return h

    def __nonzero__(self):
        """
        Check whether this element is not zero.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2+163)
            sage: not a
            False
            sage: not (a-a)
            True
        """
        return mpz_cmp_ui(self.a, 0) != 0 or mpz_cmp_ui(self.b, 0) != 0

    def _integer_(self, Z=None):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+163)
            sage: (a+1-a)._integer_()
            1
            sage: (a+1/2-a)._integer_()
            Traceback (most recent call last):
            ...
            TypeError: Unable to coerce 1/2 to an integer
        """
        cdef Integer res
        if mpz_cmp_ui(self.b, 0) != 0 or mpz_cmp_ui(self.denom, 1) != 0:
            raise TypeError("Unable to coerce %s to an integer" % self)
        else:
            res = Integer.__new__(Integer)
            mpz_set(res.value, self.a)
            return res

    def _rational_(self):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+163)
            sage: (a+1/2-a)._rational_()
            1/2
            sage: (a+1/2)._rational_()
            Traceback (most recent call last):
            ...
            TypeError: Unable to coerce a + 1/2 to a rational
        """
        cdef Rational res
        if mpz_cmp_ui(self.b, 0)!=0:
            raise TypeError("Unable to coerce %s to a rational" % self)
        else:
            res = <Rational>Rational.__new__(Rational)
            mpz_set(mpq_numref(res.value), self.a)
            mpz_set(mpq_denref(res.value), self.denom)
            mpq_canonicalize(res.value)
            return res

    def _algebraic_(self, parent):
        r"""
        Convert this element to an algebraic number, if possible.

        EXAMPLES::

            sage: NF.<i> = QuadraticField(-1)
            sage: QQbar(1+i)
            I + 1
            sage: NF.<sqrt3> = QuadraticField(2)
            sage: AA(sqrt3)
            1.414213562373095?
        """
        import sage.rings.qqbar as qqbar
        if (parent is qqbar.QQbar
                and list(self._parent.polynomial()) == [1, 0, 1]):
            # AlgebraicNumber.__init__ does a better job than
            # NumberFieldElement._algebraic_ in this case, but
            # QQbar._element_constructor_ calls the latter first.
            return qqbar.AlgebraicNumber(self)
        else:
            return NumberFieldElement._algebraic_(self, parent)

    cpdef bint is_one(self):
        r"""
        Check whether this number field element is `1`.

        EXAMPLES::

            sage: K = QuadraticField(-2)
            sage: K(1).is_one()
            True
            sage: K(-1).is_one()
            False
            sage: K(2).is_one()
            False
            sage: K(0).is_one()
            False
            sage: K(1/2).is_one()
            False
            sage: K.gen().is_one()
            False
        """
        return mpz_cmp_ui(self.a, 1) == 0 and \
               mpz_cmp_ui(self.b, 0) == 0 and \
               mpz_cmp_ui(self.denom, 1) == 0

    cpdef bint is_rational(self):
        r"""
        Check whether this number field element is a rational number.

        .. SEEALSO::

            - :meth:`is_integer` to test if this element is an integer
            - :meth:`is_integral` to test if this element is an algebraic integer

        EXAMPLES::

            sage: K.<sqrt3> = QuadraticField(3)
            sage: sqrt3.is_rational()
            False
            sage: (sqrt3-1/2).is_rational()
            False
            sage: K(0).is_rational()
            True
            sage: K(-12).is_rational()
            True
            sage: K(1/3).is_rational()
            True
        """
        return mpz_cmp_ui(self.b, 0) == 0

    def is_integer(self):
        r"""
        Check whether this number field element is an integer.

        .. SEEALSO::

            - :meth:`is_rational` to test if this element is a rational number
            - :meth:`is_integral` to test if this element is an algebraic integer

        EXAMPLES::

            sage: K.<sqrt3> = QuadraticField(3)
            sage: sqrt3.is_integer()
            False
            sage: (sqrt3-1/2).is_integer()
            False
            sage: K(0).is_integer()
            True
            sage: K(-12).is_integer()
            True
            sage: K(1/3).is_integer()
            False
        """
        return mpz_cmp_ui(self.b, 0) == mpz_cmp_ui(self.denom, 1) == 0

    def real(self):
        r"""
        Returns the real part of self, which is either self (if self lives
        it a totally real field) or a rational number.

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2)
            sage: sqrt2.real()
            sqrt2
            sage: K.<a> = QuadraticField(-3)
            sage: a.real()
            0
            sage: (a + 1/2).real()
            1/2
            sage: K.<a> = NumberField(x^2 + x + 1)
            sage: a.real()
            -1/2
            sage: parent(a.real())
            Rational Field
            sage: K.<i> = QuadraticField(-1)
            sage: i.real()
            0
        """
        cdef Rational res
        if mpz_sgn(self.D.value) > 0:
            return self  # totally real
        else:
            res = <Rational>Rational.__new__(Rational)
            mpz_set(mpq_numref(res.value), self.a)
            mpz_set(mpq_denref(res.value), self.denom)
            mpq_canonicalize(res.value)
            return res

    def imag(self):
        r"""
        Returns the imaginary part of self.

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2)
            sage: sqrt2.imag()
            0
            sage: parent(sqrt2.imag())
            Rational Field

            sage: K.<i> = QuadraticField(-1)
            sage: i.imag()
            1
            sage: parent(i.imag())
            Rational Field

            sage: K.<a> = NumberField(x^2 + x + 1, embedding=CDF.0)
            sage: a.imag()
            1/2*sqrt3
            sage: a.real()
            -1/2
            sage: SR(a)
            1/2*I*sqrt(3) - 1/2
            sage: bool(I*a.imag() + a.real() == a)
            True

        TESTS::

            sage: K.<a> = QuadraticField(-9, embedding=-CDF.0)
            sage: a.imag()
            -3
            sage: parent(a.imag())
            Rational Field

        Check that :trac:`22095` is fixed::

            sage: K.<a> = NumberField(x^2 + 2*x + 14, embedding=CC(-1,+3))
            sage: K13.<sqrt13> = QuadraticField(13)
            sage: K13.zero()
            0
            sage: a.imag()
            sqrt13
            sage: K13.zero()
            0
        """
        if mpz_sgn(self.D.value) > 0:
            return Rational.__new__(Rational)  # = 0
        cdef Integer negD = <Integer>Integer.__new__(Integer)
        mpz_neg(negD.value, self.D.value)
        cdef NumberFieldElement_quadratic q = <NumberFieldElement_quadratic>self._new()
        mpz_set_ui(q.b, 1)
        mpz_set_ui(q.denom, 1)
        cdef Rational res
        if mpz_cmp_ui(negD.value, 1) == 0 or mpz_perfect_square_p(negD.value):
            # D = -1 is the most common case we'll see here
            if self._parent._embedding is None:
                raise ValueError("Embedding must be specified.")
            res = <Rational>Rational.__new__(Rational)
            if mpz_cmp_ui(negD.value, 1) == 0:
                mpz_set(mpq_numref(res.value), self.b)
            else:
                mpz_sqrt(mpq_numref(res.value), negD.value)
                mpz_mul(mpq_numref(res.value), mpq_numref(res.value), self.b)
            mpz_set(mpq_denref(res.value), self.denom)
            mpq_canonicalize(res.value)
            if not self.standard_embedding:
                mpq_neg(res.value, res.value)
            return res
        else:
            # avoid circular import
            if self._parent._embedding is None:
                from .number_field import NumberField
                K = NumberField(QQ['x'].gen()**2 - negD, 'sqrt%s' % negD)
            else:
                from .number_field import QuadraticField
                K = QuadraticField(negD, 'sqrt%s' % negD)
            q = (<NumberFieldElement_quadratic> K._zero_element)._new()
            mpz_set(q.denom, self.denom)
            mpz_set_ui(q.a, 0)
            if self.standard_embedding:
                mpz_set(q.b, self.b)
            else:
                mpz_neg(q.b, self.b)
            return q

    cpdef list _coefficients(self):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+41)
            sage: a._coefficients()
            [0, 1]
            sage: K.<a> = NumberField(x^2+x+41)
            sage: a._coefficients()
            [0, 1]
            sage: b = 3*a+1/5
            sage: b._coefficients()
            [1/5, 3]
        """
        # In terms of the generator...
        cdef NumberFieldElement_quadratic gen = self.number_field().gen()  # should this be cached?
        cdef Rational const = <Rational>Rational.__new__(Rational)
        cdef Rational lin = <Rational>Rational.__new__(Rational)
        ad, bd = self.parts()
        if not self:
            return []
        if not bd:
            return [ad]
        if gen.is_sqrt_disc():
            return [ad,bd]
        else:
            alpha, beta = gen.parts()
            scale = bd/beta
            return [ad - scale*alpha, scale]

    def denominator(self):
        """
        Return the denominator of self. This is the LCM of the denominators of
        the coefficients of self, and thus it may well be `> 1` even when the
        element is an algebraic integer.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2+x+41)
            sage: a.denominator()
            1
            sage: b = (2*a+1)/6
            sage: b.denominator()
            6
            sage: K(1).denominator()
            1
            sage: K(1/2).denominator()
            2
            sage: K(0).denominator()
            1

            sage: K.<a> = NumberField(x^2 - 5)
            sage: b = (a + 1)/2
            sage: b.denominator()
            2
            sage: b.is_integral()
            True
        """
        # In terms of the generator...
        cdef NumberFieldElement_quadratic gen = self.number_field().gen()  # should this be cached?
        cdef Integer denom
        if gen.is_sqrt_disc():
            denom = Integer.__new__(Integer)
            mpz_set(denom.value, self.denom)
            return denom
        else:
            c = self._coefficients()
            if len(c) == 2:
                const, lin = c
            elif len(c) == 1:
                const = c[0]
                lin = Rational(0)
            else:
                const = lin = Rational(0)
            return const.denominator().lcm(lin.denominator())

    def numerator(self):
        """
        Return self*self.denominator().

        EXAMPLES::

            sage: K.<a> = NumberField(x^2+x+41)
            sage: b = (2*a+1)/6
            sage: b.denominator()
            6
            sage: b.numerator()
            2*a + 1
        """
        return self*self.denominator()


#########################################################
# Some things are so much easier to compute
#########################################################

    def trace(self):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+x+41)
            sage: a.trace()
            -1
            sage: a.matrix()
            [  0   1]
            [-41  -1]

        The trace is additive::

            sage: K.<a> = NumberField(x^2+7)
            sage: (a+1).trace()
            2
            sage: K(3).trace()
            6
            sage: (a+4).trace()
            8
            sage: (a/3+1).trace()
            2
        """
        # trace = 2*self.a / self.denom
        cdef Rational res = <Rational>Rational.__new__(Rational)
        if mpz_odd_p(self.denom):
            mpz_mul_2exp(mpq_numref(res.value), self.a, 1)
            mpz_set(mpq_denref(res.value), self.denom)
        else:
            mpz_set(mpq_numref(res.value), self.a)
            mpz_divexact_ui(mpq_denref(res.value), self.denom, 2)
        mpq_canonicalize(res.value)
        return res


    def norm(self, K=None):
        """
        Return the norm of self. If the second argument is None, this is the
        norm down to `\QQ`. Otherwise, return the norm down to K (which had
        better be either `\QQ` or this number field).

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-x+3)
            sage: a.norm()
            3
            sage: a.matrix()
            [ 0  1]
            [-3  1]
            sage: K.<a> = NumberField(x^2+5)
            sage: (1+a).norm()
            6

        The norm is multiplicative::

            sage: K.<a> = NumberField(x^2-3)
            sage: a.norm()
            -3
            sage: K(3).norm()
            9
            sage: (3*a).norm()
            -27

        We test that the optional argument is handled sensibly::

            sage: (3*a).norm(QQ)
            -27
            sage: (3*a).norm(K)
            3*a
            sage: (3*a).norm(CyclotomicField(3))
            Traceback (most recent call last):
            ...
            ValueError: no way to embed L into parent's base ring K

        """
        cdef Rational res = <Rational>Rational.__new__(Rational)

        if K is None or K == QQ:
        # norm = (a^2 - d b^2) / self.denom^2
            mpz_pow_ui(mpq_numref(res.value), self.a, 2)
            mpz_pow_ui(mpq_denref(res.value), self.b, 2)  # use as temp
            mpz_mul(mpq_denref(res.value), mpq_denref(res.value), self.D.value)
            mpz_sub(mpq_numref(res.value), mpq_numref(res.value), mpq_denref(res.value))
            mpz_pow_ui(mpq_denref(res.value), self.denom, 2)
            mpq_canonicalize(res.value)
            return res
        else:
            return NumberFieldElement.norm(self, K)

    def is_integral(self):
        r"""
        Return whether this element is an algebraic integer.

        TESTS::

            sage: K.<a> = QuadraticField(-1)
            sage: a.is_integral()
            True
            sage: K(1).is_integral()
            True
            sage: K(1/2).is_integral()
            False
            sage: (a/2).is_integral()
            False
            sage: ((a+1)/2).is_integral()
            False
            sage: ((a+1)/3).is_integral()
            False

            sage: K.<a> = QuadraticField(-3)
            sage: a.is_integral()
            True
            sage: K(1).is_integral()
            True
            sage: K(1/2).is_integral()
            False
            sage: (a/2).is_integral()
            False
            sage: ((a+1)/2).is_integral()
            True
            sage: ((a+1)/3).is_integral()
            False

        This works for order elements too, see :trac:`24077`::

            sage: O.<w> = EisensteinIntegers()
            sage: w.is_integral()
            True
            sage: for _ in range(20):
            ....:     assert O.random_element().is_integral()
        """
        if mpz_cmp_ui(self.denom, 1) == 0:
            return True

        # Check for an element of the form x + y sqrt(D) where x and y
        # are half-integers.
        if mpz_even_p(self.a) or mpz_even_p(self.b):
            return False
        if mpz_cmp_ui(self.denom, 2) != 0:
            return False
        # Numbers with half-integral components are integral only for
        # D = 1 mod 4
        return mpz_fdiv_ui(self.D.value, 4) == 1

    def charpoly(self, var='x'):
        r"""
        The characteristic polynomial of this element over `\QQ`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-x+13)
            sage: a.charpoly()
            x^2 - x + 13
            sage: b = 3-a/2
            sage: f = b.charpoly(); f
            x^2 - 11/2*x + 43/4
            sage: f(b)
            0
        """
        R = QQ[var]
        return R([self.norm(), -self.trace(), 1])

    def minpoly(self, var='x'):
        r"""
        The minimal polynomial of this element over `\QQ`.

        INPUT:

        -  ``var`` -- the minimal polynomial is defined over a polynomial ring
           in a variable with this name. If not specified this defaults to
           ``x``.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2+13)
            sage: a.minpoly()
            x^2 + 13
            sage: a.minpoly('T')
            T^2 + 13
            sage: (a+1/2-a).minpoly()
            x - 1/2
        """
        if self.is_rational():
            R = QQ[var]
            return R([-self._rational_(), 1])
        else:
            return self.charpoly(var)

    def __abs__(self):
        """
        EXAMPLES::

            sage: K.<a> = QuadraticField(2, 'a', embedding=-1.4142)
            sage: abs(a)    # indirect test
            -a
            sage: abs(a+1)  # indirect test
            -a - 1
            sage: abs(a+2)  # indirect test
            a + 2

            sage: K.<a> = NumberField(x^2+1, embedding=CDF.gen())
            sage: abs(a+1)
            sqrt(2)
        """
        if mpz_sgn(self.D.value) == 1:
            if self.sign() >= 0:
                return self
            return -self

        # doing that way the parent is the symbolic ring (or IntegerRing if the
        # norm of self is a square). On the other hand, it is coherent with
        # sage.rings.integer.Integer.sqrt
        return (self.real()**2 + self.imag()**2).sqrt()

    def floor(self):
        r"""
        Returns the floor of x.

        EXAMPLES::

            sage: K.<sqrt2> = QuadraticField(2,name='sqrt2')
            sage: sqrt2.floor()
            1
            sage: (-sqrt2).floor()
            -2
            sage: (13/197 + 3702/123*sqrt2).floor()
            42
            sage: (13/197-3702/123*sqrt2).floor()
            -43

        TESTS::

            sage: K2.<sqrt2> = QuadraticField(2)
            sage: K3.<sqrt3> = QuadraticField(3)
            sage: K5.<sqrt5> = QuadraticField(5)
            sage: for _ in range(100):
            ....:    a = QQ.random_element(1000,20)
            ....:    b = QQ.random_element(1000,20)
            ....:    assert floor(a+b*sqrt(2.)) == floor(a+b*sqrt2)
            ....:    assert floor(a+b*sqrt(3.)) == floor(a+b*sqrt3)
            ....:    assert floor(a+b*sqrt(5.)) == floor(a+b*sqrt5)

            sage: K = QuadraticField(-2)
            sage: l = [K(52), K(-3), K(43/12), K(-43/12)]
            sage: [x.floor() for x in l]
            [52, -3, 3, -4]
        """
        cdef mpz_t x
        cdef Integer result

        if mpz_sgn(self.b) == 0:
            mpz_init_set(x,self.a)
            mpz_fdiv_q(x,x,self.denom)
            result = Integer.__new__(Integer)
            mpz_set(result.value,x)
            mpz_clear(x)
            return result

        if not mpz_sgn(self.D.value) == 1:
            raise ValueError("floor is not defined for complex quadratic number field")

        mpz_init(x)
        mpz_mul(x,self.b,self.b)
        mpz_mul(x,x,self.D.value)
        mpz_sqrt(x,x)
        if mpz_sgn(self.b) == -1:
            if self.standard_embedding:
                mpz_neg(x,x)
                mpz_sub_ui(x,x,1)
        elif not self.standard_embedding:
                mpz_neg(x,x)
                mpz_sub_ui(x,x,1)

        mpz_add(x,x,self.a)    # here x = a + floor(sqrt(b^2 D)) or a + floor(-sqrt(b^2 D))
        mpz_fdiv_q(x,x,self.denom)
        result = Integer.__new__(Integer)
        mpz_set(result.value,x)
        mpz_clear(x)
        return result

    def ceil(self):
        r"""
        Returns the ceil.

        EXAMPLES::

            sage: K.<sqrt7> = QuadraticField(7, name='sqrt7')
            sage: sqrt7.ceil()
            3
            sage: (-sqrt7).ceil()
            -2
            sage: (1022/313*sqrt7 - 14/23).ceil()
            9

        TESTS::

            sage: K2.<sqrt2> = QuadraticField(2)
            sage: K3.<sqrt3> = QuadraticField(3)
            sage: K5.<sqrt5> = QuadraticField(5)
            sage: for _ in range(100):
            ....:    a = QQ.random_element(1000,20)
            ....:    b = QQ.random_element(1000,20)
            ....:    assert ceil(a+b*sqrt(2.)) == ceil(a+b*sqrt2)
            ....:    assert ceil(a+b*sqrt(3.)) == ceil(a+b*sqrt3)
            ....:    assert ceil(a+b*sqrt(5.)) == ceil(a+b*sqrt5)

            sage: K = QuadraticField(-2)
            sage: l = [K(52), K(-3), K(43/12), K(-43/12)]
            sage: [x.ceil() for x in l]
            [52, -3, 4, -3]
        """
        x = self.floor()
        if mpz_sgn(self.b) == 0 and mpz_cmp_ui(self.denom,1) == 0:
            return x
        return x+1

    def round(self):
        r"""
        Returns the round (nearest integer).

        EXAMPLES::

            sage: K.<sqrt7> = QuadraticField(7, name='sqrt7')
            sage: sqrt7.round()
            3
            sage: (-sqrt7).round()
            -3
            sage: (12/313*sqrt7 - 1745917/2902921).round()
            0
            sage: (12/313*sqrt7 - 1745918/2902921).round()
            -1

        TESTS::

            sage: K2.<sqrt2> = QuadraticField(2)
            sage: K3.<sqrt3> = QuadraticField(3)
            sage: K5.<sqrt5> = QuadraticField(5)
            sage: for _ in range(100):
            ....:    a = QQ.random_element(1000,20)
            ....:    b = QQ.random_element(1000,20)
            ....:    assert round(a) == round(K2(a)), a
            ....:    assert round(a) == round(K3(a)), a
            ....:    assert round(a) == round(K5(a)), a
            ....:    assert round(a+b*sqrt(2.)) == round(a+b*sqrt2), (a, b)
            ....:    assert round(a+b*sqrt(3.)) == round(a+b*sqrt3), (a, b)
            ....:    assert round(a+b*sqrt(5.)) == round(a+b*sqrt5), (a, b)
        """
        n = self.floor()
        test = 2 * (self - n).abs()
        if test < 1:
            return n
        elif test > 1:
            return n + 1
        elif self > 0:
            return n + 1
        else:
            return n

cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
    """
    Element of an order in a quadratic field.

    EXAMPLES::

        sage: K.<a> = NumberField(x^2 + 1)
        sage: O2 = K.order(2*a)
        sage: w = O2.1; w
        2*a
        sage: parent(w)
        Order in Number Field in a with defining polynomial x^2 + 1
    """
    def __init__(self, order, f):
        r"""
        Standard initialisation function.

        EXAMPLES::

            sage: OK.<y> = EquationOrder(x^2 + 5)
            sage: v = OK.1 # indirect doctest
            sage: type(v)
            <type 'sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic'>
        """
        K = order.number_field()
        NumberFieldElement_quadratic.__init__(self, K, f)
        (<Element>self)._parent = order

    def norm(self):
        """
        The norm of an element of the ring of integers is an Integer.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 3)
            sage: O2 = K.order(2*a)
            sage: w = O2.gen(1); w
            2*a
            sage: w.norm()
            12
            sage: parent(w.norm())
            Integer Ring
        """
        return ZZ(NumberFieldElement_quadratic.norm(self))

    def trace(self):
        """
        The trace of an element of the ring of integers is an Integer.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 - 5)
            sage: R = K.ring_of_integers()
            sage: b = R((1+a)/2)
            sage: b.trace()
            1
            sage: parent(b.trace())
            Integer Ring
        """
        return ZZ(NumberFieldElement_quadratic.trace(self))

    def charpoly(self, var='x'):
        r"""
        The characteristic polynomial of this element, which is over `\ZZ`
        because this element is an algebraic integer.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 - 5)
            sage: R = K.ring_of_integers()
            sage: b = R((5+a)/2)
            sage: f = b.charpoly('x'); f
            x^2 - 5*x + 5
            sage: f.parent()
            Univariate Polynomial Ring in x over Integer Ring
            sage: f(b)
            0
        """
        R = ZZ[var]
        return R([self.norm(), -self.trace(), 1])

    def minpoly(self, var='x'):
        r"""
        The minimal polynomial of this element over `\ZZ`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 163)
            sage: R = K.ring_of_integers()
            sage: f = R(a).minpoly('x'); f
            x^2 + 163
            sage: f.parent()
            Univariate Polynomial Ring in x over Integer Ring
            sage: R(5).minpoly()
            x - 5
        """
        if self.is_rational():
            R = ZZ[var]
            return R([-self._rational_(), 1])
        else:
            return self.charpoly()

    cdef number_field(self):
        # So few functions actually use self.number_field() for quadratic elements, so
        # it is better *not* to return a cached value (since the call to _parent.number_field())
        # is expensive.
        return self._parent.number_field()

    # We must override these since the basering is now ZZ not QQ.
    cpdef _rmul_(self, Element _c):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2-27)
            sage: R = K.ring_of_integers()
            sage: aa = R.gen(1); aa
            1/3*a
            sage: 5 * aa # indirect doctest
            5/3*a
        """
        cdef Integer c = <Integer>_c
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_mul(res.a, self.a, c.value)
        mpz_mul(res.b, self.b, c.value)
        mpz_set(res.denom, self.denom)
        res._reduce_c_()
        return res

    cpdef _lmul_(self, Element _c):
        """
        EXAMPLES::

            sage: K.<a> = NumberField(x^2+43)
            sage: R = K.ring_of_integers()
            sage: aa = R.gen(0); aa
            1/2*a + 1/2
            sage: aa*3 # indirect doctest
            3/2*a + 3/2
        """
        cdef Integer c = <Integer>_c
        cdef NumberFieldElement_quadratic res = <NumberFieldElement_quadratic>self._new()
        mpz_mul(res.a, self.a, c.value)
        mpz_mul(res.b, self.b, c.value)
        mpz_set(res.denom, self.denom)
        res._reduce_c_()
        return res

    def __invert__(self):
        r"""
        Implement inversion, checking that the return value has the right parent.
        See trac \#4190.

        EXAMPLES::

            sage: K = NumberField(x^2 -x + 2, 'a')
            sage: OK = K.ring_of_integers()
            sage: a = OK(K.gen())
            sage: (~a).parent() is K
            True
            sage: (~a) in OK
            False
            sage: a**(-1) in OK
            False
        """
        return self._parent.number_field()(NumberFieldElement_quadratic.__invert__(self))

    def inverse_mod(self, I):
        r"""
        Return an inverse of self modulo the given ideal.

        INPUT:


        -  ``I`` - may be an ideal of self.parent(), or an
           element or list of elements of self.parent() generating a nonzero
           ideal. A ValueError is raised if I is non-integral or is zero.
           A ZeroDivisionError is raised if I + (x) != (1).


        EXAMPLES::

            sage: OE.<w> = EquationOrder(x^2 - x + 2)
            sage: w.inverse_mod(13) == 6*w - 6
            True
            sage: w*(6*w - 6) - 1
            -13
            sage: w.inverse_mod(13).parent() == OE
            True
            sage: w.inverse_mod(2*OE)
            Traceback (most recent call last):
            ...
            ZeroDivisionError: w is not invertible modulo Fractional ideal (2)
        """
        R = self.parent()
        return R(_inverse_mod_generic(self, I))

cdef class Z_to_quadratic_field_element(Morphism):
    """
    Morphism that coerces from integers to elements of a quadratic number
    field K.
    """
    cdef NumberFieldElement_quadratic zero_element    # the zero element of K

    # TODO: implement __cmp__ properly so we can have a loads/dumps doctest

    def __init__(self, K):
        """
        ``K`` is the target quadratic field

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(ZZ) # indirect doctest
            sage: type(phi)
            <type 'sage.rings.number_field.number_field_element_quadratic.Z_to_quadratic_field_element'>
            sage: phi == loads(dumps(phi)) # todo: comparison not implemented
            True

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(ZZ) # indirect doctest
            sage: type(psi)
            <type 'sage.rings.number_field.number_field_element_quadratic.Z_to_quadratic_field_element'>
            sage: psi == loads(dumps(psi)) # todo: comparison not implemented
            True
        """
        import sage.categories.homset
        Morphism.__init__(self, sage.categories.homset.Hom(ZZ, K))
        self.zero_element = K.zero()

    cdef dict _extra_slots(self, dict _slots):
        """
        Helper for pickling and copying.

        TESTS::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(ZZ) # indirect doctest
            sage: phi(4)
            4
            sage: phi(5).parent() is K
            True

        """
        _slots['zero_element'] = self.zero_element
        return Morphism._extra_slots(self, _slots)

    cdef _update_slots(self, dict _slots):
        """
        Helper for pickling and copying.

        TESTS::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(ZZ) # indirect doctest
            sage: phi(4)
            4
            sage: phi(5).parent() is K
            True

        """
        Morphism._update_slots(self, _slots)
        self.zero_element = _slots['zero_element']

    cpdef Element _call_(self, x):
        r"""
        Evaluate at an integer ``x``.

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(ZZ)
            sage: a = phi(2); a # indirect doctest
            2
            sage: a.parent() is K
            True

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(ZZ)
            sage: b = psi(-42); b # indirect doctest
            -42
            sage: b.parent() is R
            True
        """
        cdef NumberFieldElement_quadratic y
        if mpz_sgn((<Integer> x).value) == 0:
            return self.zero_element
        y = self.zero_element._new()
        mpz_set(y.a, (<Integer> x).value)

        # we need to set the denominator to 1 as it is 0 for
        # the zero element of K... (because gcd(0,0) = 0).
        mpz_set_ui(y.denom, 1)

        return y

    def _repr_type(self):
        r"""
        Return a short name for this morphism.

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(ZZ)
            sage: phi # indirect doctest
            Natural morphism:
              From: Integer Ring
              To:   Number Field in a with defining polynomial x^2 - 3

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(ZZ)
            sage: psi # indirect doctest
            Natural morphism:
              From: Integer Ring
              To:   Cyclotomic Field of order 6 and degree 2
        """
        return "Natural"

cdef class Q_to_quadratic_field_element(Morphism):
    """
    Morphism that coerces from rationals to elements of a quadratic number
    field K.
    """
    cdef NumberFieldElement_quadratic zero_element    # the zero element of K

    # TODO: implement __cmp__ properly so we can have a loads/dumps doctest

    def __init__(self, K):
        """
        ``K`` is the target quadratic field

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(QQ) # indirect doctest
            sage: type(phi)
            <type 'sage.rings.number_field.number_field_element_quadratic.Q_to_quadratic_field_element'>
            sage: phi == loads(dumps(phi))  # todo: comparison not implemented
            True

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(QQ)
            sage: type(psi)
            <type 'sage.rings.number_field.number_field_element_quadratic.Q_to_quadratic_field_element'>
            sage: psi == loads(dumps(psi))  # todo: comparison not implemented
            True
        """
        import sage.categories.homset
        Morphism.__init__(self, sage.categories.homset.Hom(QQ, K))
        self.zero_element = K.zero()

    cdef dict _extra_slots(self, dict _slots):
        """
        Helper for pickling and copying.

        TESTS::

            sage: K.<a> = QuadraticField(-3) ## line 8983 ##
            sage: f = K.coerce_map_from(QQ); f # indirect doctest
            Natural morphism:
              From: Rational Field
              To:   Number Field in a with defining polynomial x^2 + 3
            sage: f(3/1)
            3
            sage: f(1/2).parent() is K
            True

        """
        _slots['zero_element'] = self.zero_element
        return Morphism._extra_slots(self, _slots)

    cdef _update_slots(self, dict _slots):
        """
        Helper for pickling and copying.

        TESTS::

            sage: K.<a> = QuadraticField(-3) ## line 8983 ##
            sage: f = K.coerce_map_from(QQ); f # indirect doctest
            Natural morphism:
              From: Rational Field
              To:   Number Field in a with defining polynomial x^2 + 3
            sage: f(3/1)
            3
            sage: f(1/2).parent() is K
            True

        """
        Morphism._update_slots(self, _slots)
        self.zero_element = _slots['zero_element']

    cpdef Element _call_(self, x):
        r"""
        Evaluate at a rational ``x``.

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(QQ)
            sage: a = phi(2/3); a # indirect doctest
            2/3
            sage: a.parent() is K
            True

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(QQ)
            sage: b = psi(-23/15); b # indirect doctest
            -23/15
            sage: b.parent() is R
            True
        """
        cdef NumberFieldElement_quadratic y = self.zero_element._new()
        mpz_set(y.a, mpq_numref((<Rational>x).value))
        mpz_set(y.denom, mpq_denref((<Rational>x).value))
        return y

    def _repr_type(self):
        r"""
        Return a short name for this morphism.

        EXAMPLES::

            sage: K.<a> = QuadraticField(3)
            sage: phi = K.coerce_map_from(QQ)
            sage: phi # indirect doctest
            Natural morphism:
              From: Rational Field
              To:   Number Field in a with defining polynomial x^2 - 3

            sage: R.<b> = CyclotomicField(6)
            sage: psi = R.coerce_map_from(QQ)
            sage: psi # indirect doctest
            Natural morphism:
              From: Rational Field
              To:   Cyclotomic Field of order 6 and degree 2
        """
        return "Natural"
