2016-01-13-part2-absolute-vs-relative.sagews

Absolute versus Relative Number Fields

William Stein

Jan 13, 2016

Let's build a tour of number fields

R1.<x> = QQ[]
F1.<a> = NumberField(x^3 + 2*x + 1)
R2.<y> = F1[]
F2.<b> = NumberField(y^2 - a)
R3.<z> = F2[]
F3.<c> = NumberField(z^3 - b)
F3

Number Field in c with defining polynomial z^3 - b over its base field

c^3

b

F2

Number Field in b with defining polynomial y^2 - a over its base field

b^2

a

F1

Number Field in a with defining polynomial x^3 + 2*x + 1

a^3

-2*a - 1

(a+b+c)^3

(3*b + 3*a)*c^2 + (6*a*b + 3*a^2 + 3*a)*c + (3*a^2 + a + 1)*b + 3*a^2 - 2*a - 1

c._add_??

   File: /projects/sage/sage-6.10/src/sage/rings/number_field/number_field_element.pyx
   Source:
       cpdef ModuleElement _add_(self, ModuleElement right):
        r"""
        EXAMPLE::

            sage: K.<s> = QuadraticField(2)
            sage: s + s # indirect doctest
            2*s
            sage: s + ZZ(3) # indirect doctest
            s + 3
        """
        cdef NumberFieldElement x
        cdef NumberFieldElement _right = right
        x = self._new()
        ZZ_mul(x.__denominator, self.__denominator, _right.__denominator)
        cdef ZZX_c t1, t2
        ZZX_mul_ZZ(t1, self.__numerator, _right.__denominator)
        ZZX_mul_ZZ(t2, _right.__numerator, self.__denominator)
        ZZX_add(x.__numerator, t1, t2)
        x._reduce_c_()
        return x

Type

open /projects/sage/sage-6.10/src/sage/rings/number_field/number_field_element.pyx

in a file term.term or browse to that file on Github:

https://github.com/sagemath/sage/blob/master/src/sage/rings/number_field/number_field_element.pyx

I also put a copy of number_field_element.pyx in our lecture notes directory here. Also see the corresponding pxd file, which defines the *data* of a number field element.

smc.link("number_field_element.pyx")
print
smc.link("number_field_element.pxd")

unknown message type 'obj'

unknown message type 'obj'

In SMC you can select all text (in number_field_element.pyx), then type Control+Q to fold all code. This gives you a nice top-down view of the file.

Look at the file.

All number field elements are represented in terms of a single absolute polynomial over $\ZZ$ and a denominator using the NTL C++ number theory library.

This is simple, but it means that in some cases arithmetic can be stupidly slow compared to Magma (and in others, very fast).

c.absolute_minpoly()

x^18 + 2*x^6 + 1

L.<alpha> = F3.absolute_field()
L

Number Field in alpha with defining polynomial x^18 + 2*x^6 + 1

The structure method allows you to recover an explicit isomorphisms between $L$ and $F_3$:

L.structure?

File: /projects/sage/sage-6.10/src/sage/rings/number_field/number_field.py
Signature : L.structure()
Docstring :
Return fixed isomorphism or embedding structure on self.

This is used to record various isomorphisms or embeddings that
arise naturally in other constructions.

EXAMPLES:

   sage: K.<z> = NumberField(x^2 + 3)
   sage: L.<a> = K.absolute_field(); L
   Number Field in a with defining polynomial x^2 + 3
   sage: L.structure()
   (Isomorphism given by variable name change map:
     From: Number Field in a with defining polynomial x^2 + 3
     To:   Number Field in z with defining polynomial x^2 + 3,
    Isomorphism given by variable name change map:
     From: Number Field in z with defining polynomial x^2 + 3
     To:   Number Field in a with defining polynomial x^2 + 3)

   sage: K.<a> = QuadraticField(-3)
   sage: R.<y> = K[]
   sage: D.<x0> = K.extension(y)
   sage: D_abs.<y0> = D.absolute_field()
   sage: D_abs.structure()[0](y0)
   -a

from_L, to_L = L.structure()

from_L(alpha + 7)

c + 7

to_L(a + b + c)

alpha^6 + alpha^3 + alpha

WARNING: use these morphisms! Do not get lazy and just try turning things into lists and converting them (copying and pasting). It'll lead to subtle bugs.

%time L.class_number(proof=False)

1 CPU time: 0.01 s, Wall time: 0.00 s

%time F3.class_number(proof=False)

1 CPU time: 0.06 s, Wall time: 0.06 s