### How to extend ring homomorphism to polynomial ring (or its fraction field)

I have a homomorphism from a number field `nf`

to the field of algebraic numbers `QQbar`

:

`nf, alpha, hom = QQbar(sqrt(2)).as_number_field_element()`

I now work in the polynomial ring `R`

over `nf`

:

`R.<x> = nf[]`

`f = x - alpha; f`

How do I get a homomorphism from `R`

to the polynomial ring over `QQbar`

extending `hom`

?
For the moment, I can use

`f.map_coefficients(hom)`

Same question about the fraction field of `R`

, e.g.,

`g = f/(x+1)`

Is there a more elegant way than calling

`g.numerator().map_coefficients(hom)/g.denominator().map_coefficients(hom)`

So basically, I'd like to extend my homomorphism `hom`

to the polynomial ring and its field of fractions.