Re: polynomial algebra question

Almost a counterexample:

take F to be the reals and A:=F[X,Y]/(X^2+Y^2+1).
Then A is no polynomial ring, because it possesses
no maximal ideal M with A/M = F.
Take K to be the complex numbers.
Then A\tensor K=K[X,Y]/(X^2+Y^2+1).

Now K[X,Y]=K[X+iY,X-iY] and X^2+Y^2+1=(X+iY)(X-iY)+1.
Therefore A\tensor K = K[X+iY]_(X+iY), that is it
is a localization of a polynomial ring.

H
.