polynomial algebra question

Suppose K/F is a field extension, A is a commutative F-algebra, and
(K tensor_F A) is isomorphic to K[x]. Must A be isomorphic to F[x]?

I know that the answer is yes with some added hypotheses, but I'm
wondering if it's true without them.

--
J. H. Palmieri
Associate Professor of Mathematics University of Washington
Box 354350, Seattle, WA 98195-4350 palmieri@xxxxxxxxxxxxxxxxxxx
http://www.math.washington.edu/~palmieri/
.

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