  Let $k$ be an uncountable algebraically closed field and let $A$ be a
countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k
K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all
subfields of the quotient division algebra of a countably generated left
Noetherian domain over $k$ are finitely generated extensions of $k$. We give
examples which show that $A\otimes_k K$ need not remain left Noetherian if the
hypotheses are weakened.
