Convert element from fraction field of polynomial ring to number field
Consider the following
F.<u> = NumberField(x^2-3)
R.<y> = PolynomialRing(F)
Q = R.fraction_field()
Then, F(Q(2*u))
yields an error since
TypeError: unable to convert 2*u to Number Field in u with defining polynomial x^2 - 3
Of course one could expect that this conversion should be not a problem. Doing the same over the base field QQ
instead of a number field works as expected.
This problem can be fixed (in this case) by converting first to the polynomial ring and then to the number field, i.e., F(R(Q(2*u)))
works fine. However, in practice, if we want to convert some a
(where we know it "should" be in F
but it might technically not) to F
, it is very unpractical to check first whether a
belongs in some certain ring and then convert it by going via the polynomial ring.
Is there a good built in way to do this? So we are given some a
(which might be already in F
or in some construction built upon F
) and want to have a
in F
.
Good question. You may want to open a trac ticket for this. It should be a matter of adding an extra case to the _convert_non_number_field_element method of NumberField.
Instead of going via
_convert_non_number_field_element
I would rather stick in a converter on the fraction field element following https://trac.sagemath.org/ticket/10147. See the fix I propose in https://trac.sagemath.org/ticket/31716.