coercion between polynomial rings over extension fields and polynomial rings over the prime subfield

[malb]

At #5569 William wrote:

As a challenge to Martin -- can you improve Sage so this decimal
string conversion (which could be a killer show stopper if the
ideal had huge elements) isn't needed, and instead one can use a
homomorphism?

The situation William is talking about is this:

sage: K.<a> = GF(2^3)
sage: P.<x,y> = PolynomialRing(K)
sage: R = PolynomialRing(GF(2),3,'a,x,y')

and we are looking for a way to convert elements in P to elements in R.

CC: @williamstein

Component: commutative algebra

Keywords: polynomial ring

Issue created by migration from https://trac.sagemath.org/ticket/5590