Elements optimized for quadratic number fields

This module defines a Cython class NumberFieldElement_quadratic to speed up computations in quadratic extensions of \(\QQ\).

Todo

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

AUTHORS:

  • Robert Bradshaw (2007-09): initial version

  • David Harvey (2007-10): fixed up a few bugs, polish around the edges

  • David Loeffler (2009-05): added more documentation and tests

  • Vincent Delecroix (2012-07): added comparisons for quadratic number fields (Issue #13213), abs, floor and ceil functions (Issue #13256)

class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian[source]

Bases: NumberFieldElement_quadratic_sqrt

An element of \(\QQ[i]\).

Some methods of this class behave slightly differently than the corresponding methods of general elements of quadratic number fields, especially with regard to conversions to parents that can represent complex numbers in rectangular form.

In addition, this class provides some convenience methods similar to methods of symbolic expressions to make the behavior of a + I*b with rational a, b closer to that when a, b are expressions.

EXAMPLES:

sage: type(I)
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian'>

sage: mi = QuadraticField(-1, embedding=CC(0,-1)).gen()
sage: type(mi)
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian'>
sage: CC(mi)
-1.00000000000000*I

>>> from sage.all import *
>>> type(I)
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian'>

>>> mi = QuadraticField(-Integer(1), embedding=CC(Integer(0),-Integer(1))).gen()
>>> type(mi)
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian'>
>>> CC(mi)
-1.00000000000000*I

type(I)
mi = QuadraticField(-1, embedding=CC(0,-1)).gen()
type(mi)
CC(mi)
imag()[source]

Imaginary part.

EXAMPLES:

sage: (1 + 2*I).imag()
2
sage: (1 + 2*I).imag().parent()
Rational Field

sage: K.<mi> = QuadraticField(-1, embedding=CC(0,-1))
sage: (1 - mi).imag()
1

>>> from sage.all import *
>>> (Integer(1) + Integer(2)*I).imag()
2
>>> (Integer(1) + Integer(2)*I).imag().parent()
Rational Field

>>> K = QuadraticField(-Integer(1), embedding=CC(Integer(0),-Integer(1)), names=('mi',)); (mi,) = K._first_ngens(1)
>>> (Integer(1) - mi).imag()
1

(1 + 2*I).imag()
(1 + 2*I).imag().parent()
K.<mi> = QuadraticField(-1, embedding=CC(0,-1))
(1 - mi).imag()
imag_part()[source]

Imaginary part.

EXAMPLES:

sage: (1 + 2*I).imag()
2
sage: (1 + 2*I).imag().parent()
Rational Field

sage: K.<mi> = QuadraticField(-1, embedding=CC(0,-1))
sage: (1 - mi).imag()
1

>>> from sage.all import *
>>> (Integer(1) + Integer(2)*I).imag()
2
>>> (Integer(1) + Integer(2)*I).imag().parent()
Rational Field

>>> K = QuadraticField(-Integer(1), embedding=CC(Integer(0),-Integer(1)), names=('mi',)); (mi,) = K._first_ngens(1)
>>> (Integer(1) - mi).imag()
1

(1 + 2*I).imag()
(1 + 2*I).imag().parent()
K.<mi> = QuadraticField(-1, embedding=CC(0,-1))
(1 - mi).imag()
log(*args, **kwds)[source]

Complex logarithm (standard branch).

EXAMPLES:

sage: I.log()                                                               # needs sage.symbolic
1/2*I*pi

>>> from sage.all import *
>>> I.log()                                                               # needs sage.symbolic
1/2*I*pi

I.log()                                                               # needs sage.symbolic
real()[source]

Real part.

EXAMPLES:

sage: (1 + 2*I).real()
1
sage: (1 + 2*I).real().parent()
Rational Field

>>> from sage.all import *
>>> (Integer(1) + Integer(2)*I).real()
1
>>> (Integer(1) + Integer(2)*I).real().parent()
Rational Field

(1 + 2*I).real()
(1 + 2*I).real().parent()
real_part()[source]

Real part.

EXAMPLES:

sage: (1 + 2*I).real()
1
sage: (1 + 2*I).real().parent()
Rational Field

>>> from sage.all import *
>>> (Integer(1) + Integer(2)*I).real()
1
>>> (Integer(1) + Integer(2)*I).real().parent()
Rational Field

(1 + 2*I).real()
(1 + 2*I).real().parent()
class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic[source]

Bases: NumberFieldElement_absolute

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 \(\gcd(a, b, c) = 1\) 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 \(c = 1\).

ceil()[source]

Return 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

>>> from sage.all import *
>>> K = QuadraticField(Integer(7), name='sqrt7', names=('sqrt7',)); (sqrt7,) = K._first_ngens(1)
>>> sqrt7.ceil()
3
>>> (-sqrt7).ceil()
-2
>>> (Integer(1022)/Integer(313)*sqrt7 - Integer(14)/Integer(23)).ceil()
9

K.<sqrt7> = QuadraticField(7, name='sqrt7')
sqrt7.ceil()
(-sqrt7).ceil()
(1022/313*sqrt7 - 14/23).ceil()
charpoly(var='x', algorithm=None)[source]

The characteristic 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'

  • algorithm – for compatibility with general number field elements; ignored

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - x + Integer(13), names=('a',)); (a,) = K._first_ngens(1)
>>> a.charpoly()
x^2 - x + 13
>>> b = Integer(3) - a/Integer(2)
>>> f = b.charpoly(); f
x^2 - 11/2*x + 43/4
>>> f(b)
0

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - x + 13)
a.charpoly()
b = 3 - a/2
f = b.charpoly(); f
f(b)
continued_fraction()[source]

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)*]

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> cf = sqrt2.continued_fraction(); cf
[1; (2)*]
>>> cf.n()
1.41421356237310
>>> sqrt2.n()
1.41421356237309
>>> cf.value()
sqrt2

>>> (sqrt2/Integer(3) + Integer(1)/Integer(4)).continued_fraction()
[0; 1, (2, 1, 1, 2, 3, 2, 1, 1, 2, 5, 1, 1, 14, 1, 1, 5)*]

K.<sqrt2> = QuadraticField(2)
cf = sqrt2.continued_fraction(); cf
cf.n()
sqrt2.n()
cf.value()
(sqrt2/3 + 1/4).continued_fraction()
continued_fraction_list()[source]

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))

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> sqrt2.continued_fraction_list()
((1,), (2,))
>>> (Integer(1)/Integer(2) + sqrt2/Integer(3)).continued_fraction_list()
((0, 1, 33), (1, 32))

K.<sqrt2> = QuadraticField(2)
sqrt2.continued_fraction_list()
(1/2 + sqrt2/3).continued_fraction_list()

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), ())

>>> from sage.all import *
>>> K(Integer(123)/Integer(567)).continued_fraction_list()
((0, 4, 1, 1, 1, 1, 3, 2), ())

K(123/567).continued_fraction_list()
denominator()[source]

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: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 - 5)
sage: b = (a + 1)/2
sage: b.denominator()
2
sage: b.is_integral()
True

sage: K.<c> = NumberField(x^2 - x + 7)
sage: c.denominator()
1

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> b = (a + Integer(1))/Integer(2)
>>> b.denominator()
2
>>> b.is_integral()
True

>>> K = NumberField(x**Integer(2) - x + Integer(7), names=('c',)); (c,) = K._first_ngens(1)
>>> c.denominator()
1

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 5)
b = (a + 1)/2
b.denominator()
b.is_integral()
K.<c> = NumberField(x^2 - x + 7)
c.denominator()
floor()[source]

Return the floor of self.

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

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), name='sqrt2', names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> sqrt2.floor()
1
>>> (-sqrt2).floor()
-2
>>> (Integer(13)/Integer(197) + Integer(3702)/Integer(123)*sqrt2).floor()
42
>>> (Integer(13)/Integer(197) - Integer(3702)/Integer(123)*sqrt2).floor()
-43

K.<sqrt2> = QuadraticField(2, name='sqrt2')
sqrt2.floor()
(-sqrt2).floor()
(13/197 + 3702/123*sqrt2).floor()
(13/197 - 3702/123*sqrt2).floor()
galois_conjugate()[source]

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: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> K = QuadraticField(Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> a.galois_conjugate()
-a

>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(5)*x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> a.galois_conjugate()
-a + 5
>>> b = Integer(5)*a + Integer(1)/Integer(3)
>>> b.galois_conjugate()
-5*a + 76/3
>>> b.norm() ==  b * b.galois_conjugate()
True
>>> b.trace() ==  b + b.galois_conjugate()
True

K.<a> = QuadraticField(23)
a.galois_conjugate()
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 5*x + 1)
a.galois_conjugate()
b = 5*a + 1/3
b.galois_conjugate()
b.norm() ==  b * b.galois_conjugate()
b.trace() ==  b + b.galois_conjugate()
imag()[source]

Return 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: x = polygen(ZZ, 'x')
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)                                                                 # needs sage.symbolic
1/2*I*sqrt(3) - 1/2
sage: bool(QQbar(I)*QQbar(a.imag()) + QQbar(a.real()) == QQbar(a))
True

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> sqrt2.imag()
0
>>> parent(sqrt2.imag())
Rational Field

>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> i.imag()
1
>>> parent(i.imag())
Rational Field

>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(1), embedding=CDF.gen(0), names=('a',)); (a,) = K._first_ngens(1)
>>> a.imag()
1/2*sqrt3
>>> a.real()
-1/2
>>> SR(a)                                                                 # needs sage.symbolic
1/2*I*sqrt(3) - 1/2
>>> bool(QQbar(I)*QQbar(a.imag()) + QQbar(a.real()) == QQbar(a))
True

K.<sqrt2> = QuadraticField(2)
sqrt2.imag()
parent(sqrt2.imag())
K.<i> = QuadraticField(-1)
i.imag()
parent(i.imag())
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + x + 1, embedding=CDF.0)
a.imag()
a.real()
SR(a)                                                                 # needs sage.symbolic
bool(QQbar(I)*QQbar(a.imag()) + QQbar(a.real()) == QQbar(a))
is_integer()[source]

Check whether this number field element is an integer.

See also

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

>>> from sage.all import *
>>> K = QuadraticField(Integer(3), names=('sqrt3',)); (sqrt3,) = K._first_ngens(1)
>>> sqrt3.is_integer()
False
>>> (sqrt3 - Integer(1)/Integer(2)).is_integer()
False
>>> K(Integer(0)).is_integer()
True
>>> K(-Integer(12)).is_integer()
True
>>> K(Integer(1)/Integer(3)).is_integer()
False

K.<sqrt3> = QuadraticField(3)
sqrt3.is_integer()
(sqrt3 - 1/2).is_integer()
K(0).is_integer()
K(-12).is_integer()
K(1/3).is_integer()
is_integral()[source]

Return whether this element is an algebraic integer.

is_one()[source]

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

>>> from sage.all import *
>>> K = QuadraticField(-Integer(2))
>>> K(Integer(1)).is_one()
True
>>> K(-Integer(1)).is_one()
False
>>> K(Integer(2)).is_one()
False
>>> K(Integer(0)).is_one()
False
>>> K(Integer(1)/Integer(2)).is_one()
False
>>> K.gen().is_one()
False

K = QuadraticField(-2)
K(1).is_one()
K(-1).is_one()
K(2).is_one()
K(0).is_one()
K(1/2).is_one()
K.gen().is_one()
is_rational()[source]

Check whether this number field element is a rational number.

See also

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

>>> from sage.all import *
>>> K = QuadraticField(Integer(3), names=('sqrt3',)); (sqrt3,) = K._first_ngens(1)
>>> sqrt3.is_rational()
False
>>> (sqrt3 - Integer(1)/Integer(2)).is_rational()
False
>>> K(Integer(0)).is_rational()
True
>>> K(-Integer(12)).is_rational()
True
>>> K(Integer(1)/Integer(3)).is_rational()
True

K.<sqrt3> = QuadraticField(3)
sqrt3.is_rational()
(sqrt3 - 1/2).is_rational()
K(0).is_rational()
K(-12).is_rational()
K(1/3).is_rational()
minpoly(var='x', algorithm=None)[source]

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'

  • algorithm – for compatibility with general number field elements; ignored

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(13), names=('a',)); (a,) = K._first_ngens(1)
>>> a.minpoly()
x^2 + 13
>>> a.minpoly('T')
T^2 + 13
>>> (a + Integer(1)/Integer(2) - a).minpoly()
x - 1/2

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + 13)
a.minpoly()
a.minpoly('T')
(a + 1/2 - a).minpoly()
norm(K=None)[source]

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: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - x + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> a.norm()
3
>>> a.matrix()
[ 0  1]
[-3  1]
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> (Integer(1) + a).norm()
6

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - x + 3)
a.norm()
a.matrix()
K.<a> = NumberField(x^2 + 5)
(1 + a).norm()

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

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> a.norm()
-3
>>> K(Integer(3)).norm()
9
>>> (Integer(3)*a).norm()
-27

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

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

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

(3*a).norm(QQ)
(3*a).norm(K)
(3*a).norm(CyclotomicField(3))
numerator()[source]

Return self * self.denominator().

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 + x + 41)
sage: b = (2*a+1)/6
sage: b.denominator()
6
sage: b.numerator()
2*a + 1

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(41), names=('a',)); (a,) = K._first_ngens(1)
>>> b = (Integer(2)*a+Integer(1))/Integer(6)
>>> b.denominator()
6
>>> b.numerator()
2*a + 1

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + x + 41)
b = (2*a+1)/6
b.denominator()
b.numerator()
parts()[source]

Return 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: x = polygen(ZZ, 'x')
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]

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(13), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant()
13
>>> a.parts()
(0, 1)
>>> (a/Integer(2) - Integer(4)).parts()
(-4, 1/2)
>>> K = NumberField(x**Integer(2) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant()
28
>>> a.parts()
(0, 1)
>>> K = NumberField(x**Integer(2) - x + Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> a.parts()
(1/2, 3/2)
>>> a._coefficients()
[0, 1]

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 13)
K.discriminant()
a.parts()
(a/2 - 4).parts()
K.<a> = NumberField(x^2 - 7)
K.discriminant()
a.parts()
K.<a> = NumberField(x^2 - x + 7)
a.parts()
a._coefficients()
real()[source]

Return the real part of self, which is either self (if self lives in 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: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> sqrt2.real()
sqrt2
>>> K = QuadraticField(-Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> a.real()
0
>>> (a + Integer(1)/Integer(2)).real()
1/2
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> a.real()
-1/2
>>> parent(a.real())
Rational Field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> i.real()
0

K.<sqrt2> = QuadraticField(2)
sqrt2.real()
K.<a> = QuadraticField(-3)
a.real()
(a + 1/2).real()
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + x + 1)
a.real()
parent(a.real())
K.<i> = QuadraticField(-1)
i.real()
round()[source]

Return the round (nearest integer) of this number field element. In case of ties, this relies on the default rounding for rational numbers.

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

>>> from sage.all import *
>>> K = QuadraticField(Integer(7), name='sqrt7', names=('sqrt7',)); (sqrt7,) = K._first_ngens(1)
>>> sqrt7.round()
3
>>> (-sqrt7).round()
-3
>>> (Integer(12)/Integer(313)*sqrt7 - Integer(1745917)/Integer(2902921)).round()
0
>>> (Integer(12)/Integer(313)*sqrt7 - Integer(1745918)/Integer(2902921)).round()
-1

K.<sqrt7> = QuadraticField(7, name='sqrt7')
sqrt7.round()
(-sqrt7).round()
(12/313*sqrt7 - 1745917/2902921).round()
(12/313*sqrt7 - 1745918/2902921).round()
sign()[source]

Return 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: # needs sage.symbolic
sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), name='sqrt2', names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> K(Integer(0)).sign()
0
>>> sqrt2.sign()
1
>>> (sqrt2+Integer(1)).sign()
1
>>> (sqrt2-Integer(1)).sign()
1
>>> (sqrt2-Integer(2)).sign()
-1
>>> (-sqrt2).sign()
-1
>>> (-sqrt2+Integer(1)).sign()
-1
>>> (-sqrt2+Integer(2)).sign()
1

>>> K = QuadraticField(Integer(2), embedding=-RealNumber('1.4142'), names=('a',)); (a,) = K._first_ngens(1)
>>> K(Integer(0)).sign()
0
>>> a.sign()
-1
>>> (a+Integer(1)).sign()
-1
>>> (a+Integer(2)).sign()
1
>>> (a-Integer(1)).sign()
-1
>>> (-a).sign()
1
>>> (-a-Integer(1)).sign()
1
>>> (-a-Integer(2)).sign()
-1

>>> # needs sage.symbolic
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(2)*x + Integer(7), 'b', embedding=CC(-Integer(1),-sqrt(Integer(6))), names=('b',)); (b,) = K._first_ngens(1)
>>> b.sign()
Traceback (most recent call last):
...
ValueError: a complex number has no sign!
>>> K(Integer(1)).sign()
1
>>> K(Integer(0)).sign()
0
>>> K(-Integer(2)/Integer(3)).sign()
-1

K.<sqrt2> = QuadraticField(2, name='sqrt2')
K(0).sign()
sqrt2.sign()
(sqrt2+1).sign()
(sqrt2-1).sign()
(sqrt2-2).sign()
(-sqrt2).sign()
(-sqrt2+1).sign()
(-sqrt2+2).sign()
K.<a> = QuadraticField(2, embedding=-1.4142)
K(0).sign()
a.sign()
(a+1).sign()
(a+2).sign()
(a-1).sign()
(-a).sign()
(-a-1).sign()
(-a-2).sign()
# needs sage.symbolic
x = polygen(ZZ, 'x')
K.<b> = NumberField(x^2 + 2*x + 7, 'b', embedding=CC(-1,-sqrt(6)))
b.sign()
K(1).sign()
K(0).sign()
K(-2/3).sign()
trace()[source]

EXAMPLES:

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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(41), names=('a',)); (a,) = K._first_ngens(1)
>>> a.trace()
-1
>>> a.matrix()
[  0   1]
[-41  -1]

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + x + 41)
a.trace()
a.matrix()

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

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> (a + Integer(1)).trace()
2
>>> K(Integer(3)).trace()
6
>>> (a + Integer(4)).trace()
8
>>> (a/Integer(3) + Integer(1)).trace()
2

K.<a> = NumberField(x^2 + 7)
(a + 1).trace()
K(3).trace()
(a + 4).trace()
(a/3 + 1).trace()
class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic_sqrt[source]

Bases: NumberFieldElement_quadratic

A NumberFieldElement_quadratic_sqrt object gives an efficient representation of an element of a quadratic extension of \(\QQ\) for the case when is_sqrt_disc() is True.

denominator()[source]

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: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(41), names=('a',)); (a,) = K._first_ngens(1)
>>> a.denominator()
1
>>> b = (Integer(2)*a+Integer(1))/Integer(6)
>>> b.denominator()
6
>>> K(Integer(1)).denominator()
1
>>> K(Integer(1)/Integer(2)).denominator()
2
>>> K(Integer(0)).denominator()
1

>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> b = (a + Integer(1))/Integer(2)
>>> b.denominator()
2
>>> b.is_integral()
True

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + x + 41)
a.denominator()
b = (2*a+1)/6
b.denominator()
K(1).denominator()
K(1/2).denominator()
K(0).denominator()
K.<a> = NumberField(x^2 - 5)
b = (a + 1)/2
b.denominator()
b.is_integral()
class sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic[source]

Bases: NumberFieldElement_quadratic

Element of an order in a quadratic field.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 + 1)
sage: O2 = K.order(2*a)
sage: w = O2.1; w
2*a
sage: parent(w)
Order of conductor 2 generated by 2*a in Number Field in a with defining polynomial x^2 + 1

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> O2 = K.order(Integer(2)*a)
>>> w = O2.gen(1); w
2*a
>>> parent(w)
Order of conductor 2 generated by 2*a in Number Field in a with defining polynomial x^2 + 1

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + 1)
O2 = K.order(2*a)
w = O2.1; w
parent(w)
charpoly(var='x', algorithm=None)[source]

The characteristic polynomial of this element, which is over \(\ZZ\) because this element is an algebraic integer.

INPUT:

  • var – the minimal polynomial is defined over a polynomial ring in a variable with this name; if not specified, this defaults to 'x'

  • algorithm – for compatibility with general number field elements; ignored

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K.ring_of_integers()
>>> b = R((Integer(5)+a)/Integer(2))
>>> f = b.charpoly('x'); f
x^2 - 5*x + 5
>>> f.parent()
Univariate Polynomial Ring in x over Integer Ring
>>> f(b)
0

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 5)
R = K.ring_of_integers()
b = R((5+a)/2)
f = b.charpoly('x'); f
f.parent()
f(b)
denominator()[source]

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: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 - 27)
sage: R = K.ring_of_integers()
sage: aa = R.gen(1)
sage: aa.denominator()
3

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(27), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K.ring_of_integers()
>>> aa = R.gen(Integer(1))
>>> aa.denominator()
3

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 27)
R = K.ring_of_integers()
aa = R.gen(1)
aa.denominator()
inverse_mod(I)[source]

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 \(I + (x) \neq (1)\).

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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)
Traceback (most recent call last):
...
ZeroDivisionError: w is not invertible modulo Fractional ideal (2)

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> OE = EquationOrder(x**Integer(2) - x + Integer(2), names=('w',)); (w,) = OE._first_ngens(1)
>>> w.inverse_mod(Integer(13)) == Integer(6)*w - Integer(6)
True
>>> w*(Integer(6)*w - Integer(6)) - Integer(1)
-13
>>> w.inverse_mod(Integer(13)).parent() == OE
True
>>> w.inverse_mod(Integer(2))
Traceback (most recent call last):
...
ZeroDivisionError: w is not invertible modulo Fractional ideal (2)

x = polygen(ZZ, 'x')
OE.<w> = EquationOrder(x^2 - x + 2)
w.inverse_mod(13) == 6*w - 6
w*(6*w - 6) - 1
w.inverse_mod(13).parent() == OE
w.inverse_mod(2)
minpoly(var='x', algorithm=None)[source]

The minimal polynomial of this element over \(\ZZ\).

INPUT:

  • var – the minimal polynomial is defined over a polynomial ring in a variable with this name; if not specified, this defaults to 'x'

  • algorithm – for compatibility with general number field elements; ignored

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(163), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K.ring_of_integers()
>>> f = R(a).minpoly('x'); f
x^2 + 163
>>> f.parent()
Univariate Polynomial Ring in x over Integer Ring
>>> R(Integer(5)).minpoly()
x - 5

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + 163)
R = K.ring_of_integers()
f = R(a).minpoly('x'); f
f.parent()
R(5).minpoly()
norm()[source]

The norm of an element of the ring of integers is an Integer.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> O2 = K.order(Integer(2)*a)
>>> w = O2.gen(Integer(1)); w
2*a
>>> w.norm()
12
>>> parent(w.norm())
Integer Ring

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + 3)
O2 = K.order(2*a)
w = O2.gen(1); w
w.norm()
parent(w.norm())
trace()[source]

The trace of an element of the ring of integers is an Integer.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K.ring_of_integers()
>>> b = R((Integer(1)+a)/Integer(2))
>>> b.trace()
1
>>> parent(b.trace())
Integer Ring

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - 5)
R = K.ring_of_integers()
b = R((1+a)/2)
b.trace()
parent(b.trace())
class sage.rings.number_field.number_field_element_quadratic.Q_to_quadratic_field_element[source]

Bases: Morphism

Morphism that coerces from rationals to elements of a quadratic number field \(K\).

EXAMPLES:

sage: K.<a> = QuadraticField(-3)
sage: f = K.coerce_map_from(QQ); f
Natural morphism:
  From: Rational Field
  To:   Number Field in a with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
sage: f(3/1)
3
sage: f(1/2).parent() is K
True

>>> from sage.all import *
>>> K = QuadraticField(-Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> f = K.coerce_map_from(QQ); f
Natural morphism:
  From: Rational Field
  To:   Number Field in a with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
>>> f(Integer(3)/Integer(1))
3
>>> f(Integer(1)/Integer(2)).parent() is K
True

K.<a> = QuadraticField(-3)
f = K.coerce_map_from(QQ); f
f(3/1)
f(1/2).parent() is K
class sage.rings.number_field.number_field_element_quadratic.Z_to_quadratic_field_element[source]

Bases: Morphism

Morphism that coerces from integers to elements of a quadratic number field \(K\).

EXAMPLES:

sage: K.<a> = QuadraticField(3)
sage: phi = K.coerce_map_from(ZZ); phi
Natural morphism:
  From: Integer Ring
  To:   Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
sage: phi(4)
4
sage: phi(5).parent() is K
True

>>> from sage.all import *
>>> K = QuadraticField(Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> phi = K.coerce_map_from(ZZ); phi
Natural morphism:
  From: Integer Ring
  To:   Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
>>> phi(Integer(4))
4
>>> phi(Integer(5)).parent() is K
True

K.<a> = QuadraticField(3)
phi = K.coerce_map_from(ZZ); phi
phi(4)
phi(5).parent() is K
sage.rings.number_field.number_field_element_quadratic.is_sqrt_disc(ad, bd)[source]

Return True if the pair (ad, bd) is \(\sqrt{D}\).

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: F.<b> = NumberField(x^2 - x + 7)
sage: b.denominator()  # indirect doctest
1

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> F = NumberField(x**Integer(2) - x + Integer(7), names=('b',)); (b,) = F._first_ngens(1)
>>> b.denominator()  # indirect doctest
1

x = polygen(ZZ, 'x')
F.<b> = NumberField(x^2 - x + 7)
b.denominator()  # indirect doctest