Noetherian algebras over algebraically closed fields

Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that A kK 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 kK need not remain left Noetherian if the hypotheses are weakened