notes_on_number_fields

In order to work with arbitrary number fields, the structure of the
number fields has to be slightly modified.

Interesting things:

- sage.rings.real_mpfi: where the real interval field RIF is defined (an
  element in that field is a pair of floating points).
- sage.rings.qqbar: where the set of algebraic real (AA) is defined.
- sage.rings.number_field.number_field: where number fields are defined.
  Note that the class for elements of quadratic fields differ slightly
  from the more general ones.
- sage.rings.polynomial.real_roots: compute isolating intervals for
  roots of a polynomial. The results might either be a pair of rationals
  or an element of RIF

Examples and problems

    sage: R.<x> = PolynomialRing(QQ)
    sage: K.<cbrt2> = NumberField(x^3-2, embedding=1.2599)

Currently the following fails

   sage: AA(cbrt2)
   Traceback (most recent call last):
   ...
   TypeError: Illegal initializer for algebraic number

For sure, defining embedding=1.2599 defines an embedding into RR which is
the set of floating point numbers!! So each time we use a number field we
must use

    sage: R.<x> = PolynomialRing(QQ)
    sage: P = x^3 - 2
    sage: cbrt2_AA = AA.polynomial_root(P, RIF(1.259,1.26))
    sage: K.<cbrt2> = NumberField(P, embedding=cbrt2_AA)

The problem with the trick is that the number field is actually built twice...
(once in the background of AA and once in the last command above).