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 (github issue #13213), abs, floor and ceil functions (github issue #13256)

class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian#

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

imag()#

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

imag_part()#

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

log(*args, **kwds)#

Complex logarithm (standard branch).

EXAMPLES:

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

real()#

Real part.

EXAMPLES:

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

real_part()#

Real part.

EXAMPLES:

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

class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic#

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()#

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

charpoly(var='x', algorithm=None)#

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

continued_fraction()#

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)*]

continued_fraction_list()#

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), ())

denominator()#

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

floor()#

Returns 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

galois_conjugate()#

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

imag()#

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

is_integer()#

Check whether this number field element is an integer.

See also

is_integer() to test if this element is an integer
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

minpoly(var='x', algorithm=None)#

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: and 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

norm(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: 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

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

numerator()#

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

parts()#

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]

real()#

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

round()#

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

sign()#

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: # 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

trace()#

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]

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

class sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic_sqrt#

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()#

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

class sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic#

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

charpoly(var='x', algorithm=None)#

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

denominator()#

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

inverse_mod(I)#

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*OE)
Traceback (most recent call last):
...
ZeroDivisionError: w is not invertible modulo Fractional ideal (2)

minpoly(var='x', algorithm=None)#

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

norm()#

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

trace()#

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

class sage.rings.number_field.number_field_element_quadratic.Q_to_quadratic_field_element#

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

class sage.rings.number_field.number_field_element_quadratic.Z_to_quadratic_field_element#

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

sage.rings.number_field.number_field_element_quadratic.is_sqrt_disc(ad, bd)#

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