Number fields

class sage.categories.number_fields.NumberFields

Bases: Category_singleton

The category of number fields.

EXAMPLES:

We create the category of number fields:

Sage

sage: C = NumberFields()
sage: C
Category of number fields

Python

>>> from sage.all import *
>>> C = NumberFields()
>>> C
Category of number fields

By definition, it is infinite:

Sage

sage: NumberFields().Infinite() is NumberFields()
True

Python

>>> from sage.all import *
>>> NumberFields().Infinite() is NumberFields()
True

Notice that the rational numbers \(\QQ\) are considered as an object in this category:

Sage

sage: RationalField() in C
True

Python

>>> from sage.all import *
>>> RationalField() in C
True

However, we can define a degree 1 extension of \(\QQ\), which is of course also in this category:

Sage

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: K = NumberField(x - 1, 'a'); K                                            # needs sage.rings.number_field
Number Field in a with defining polynomial x - 1
sage: K in C                                                                    # needs sage.rings.number_field
True

Python

>>> from sage.all import *
>>> x = PolynomialRing(RationalField(), 'x').gen()
>>> K = NumberField(x - Integer(1), 'a'); K                                            # needs sage.rings.number_field
Number Field in a with defining polynomial x - 1
>>> K in C                                                                    # needs sage.rings.number_field
True

Number fields all lie in this category, regardless of the name of the variable:

Sage

sage: K = NumberField(x^2 + 1, 'a')                                             # needs sage.rings.number_field
sage: K in C                                                                    # needs sage.rings.number_field
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), 'a')                                             # needs sage.rings.number_field
>>> K in C                                                                    # needs sage.rings.number_field
True

class ElementMethods

Bases: object

class ParentMethods

Bases: object

zeta_function(prec=53, max_imaginary_part=0, max_asymp_coeffs=40, algorithm='pari')

Return the Dedekind zeta function of this number field.

Actually, this returns an interface for computing with the Dedekind zeta function \(\zeta_F(s)\) of the number field \(F\).

INPUT:

• prec – integer (default: 53); bits of precision

• max_imaginary_part – real (default: 0)

• max_asymp_coeffs – integer (default: 40)

• algorithm – (default: 'pari') either 'gp' or 'pari'

OUTPUT: the zeta function of this number field

If algorithm is 'gp', this returns an interface to Tim Dokchitser’s gp script for computing with \(L\)-functions.

If algorithm is 'pari', this returns instead an interface to Pari’s own general implementation of \(L\)-functions.

EXAMPLES:

Sage

sage: K.<a> = NumberField(ZZ['x'].0^2 + ZZ['x'].0 - 1)                  # needs sage.rings.number_field
sage: Z = K.zeta_function(); Z                                          # needs sage.rings.number_field sage.symbolic
PARI zeta function associated to Number Field in a
 with defining polynomial x^2 + x - 1
sage: Z(-1)                                                             # needs sage.rings.number_field sage.symbolic
0.0333333333333333

sage: x = polygen(QQ, 'x')
sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1])            # needs sage.rings.number_field
sage: Z = L.zeta_function()                                             # needs sage.rings.number_field sage.symbolic
sage: Z(5)                                                              # needs sage.rings.number_field sage.symbolic
1.00199015670185

Python

>>> from sage.all import *
>>> K = NumberField(ZZ['x'].gen(0)**Integer(2) + ZZ['x'].gen(0) - Integer(1), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> Z = K.zeta_function(); Z                                          # needs sage.rings.number_field sage.symbolic
PARI zeta function associated to Number Field in a
 with defining polynomial x^2 + x - 1
>>> Z(-Integer(1))                                                             # needs sage.rings.number_field sage.symbolic
0.0333333333333333

>>> x = polygen(QQ, 'x')
>>> L = NumberField([x**Integer(2) - Integer(5), x**Integer(2) + Integer(3), x**Integer(2) + Integer(1)], names=('a', 'b', 'c',)); (a, b, c,) = L._first_ngens(3)# needs sage.rings.number_field
>>> Z = L.zeta_function()                                             # needs sage.rings.number_field sage.symbolic
>>> Z(Integer(5))                                                              # needs sage.rings.number_field sage.symbolic
1.00199015670185

Using the algorithm “pari”:

Sage

sage: K.<a> = NumberField(ZZ['x'].0^2 + ZZ['x'].0 - 1)                  # needs sage.rings.number_field
sage: Z = K.zeta_function(algorithm='pari')                             # needs sage.rings.number_field sage.symbolic
sage: Z(-1)                                                             # needs sage.rings.number_field sage.symbolic
0.0333333333333333

sage: x = polygen(QQ, 'x')
sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1])            # needs sage.rings.number_field
sage: Z = L.zeta_function(algorithm='pari')                             # needs sage.rings.number_field sage.symbolic
sage: Z(5)                                                              # needs sage.rings.number_field sage.symbolic
1.00199015670185

Python

>>> from sage.all import *
>>> K = NumberField(ZZ['x'].gen(0)**Integer(2) + ZZ['x'].gen(0) - Integer(1), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> Z = K.zeta_function(algorithm='pari')                             # needs sage.rings.number_field sage.symbolic
>>> Z(-Integer(1))                                                             # needs sage.rings.number_field sage.symbolic
0.0333333333333333

>>> x = polygen(QQ, 'x')
>>> L = NumberField([x**Integer(2) - Integer(5), x**Integer(2) + Integer(3), x**Integer(2) + Integer(1)], names=('a', 'b', 'c',)); (a, b, c,) = L._first_ngens(3)# needs sage.rings.number_field
>>> Z = L.zeta_function(algorithm='pari')                             # needs sage.rings.number_field sage.symbolic
>>> Z(Integer(5))                                                              # needs sage.rings.number_field sage.symbolic
1.00199015670185

super_categories()

EXAMPLES:

Sage

sage: NumberFields().super_categories()
[Category of infinite fields]

Python

>>> from sage.all import *
>>> NumberFields().super_categories()
[Category of infinite fields]