The question is as follows: Assume that k and F are fields and that k is finitely generated as an F-algebra. It is necessarily true that every subfield of k is finitely generated as an F algebra. I haven't been able to think of a counter example, and I can only show the result holds in special cases. Namely, if L is the subfield, I can show that L is finitely generated if (1) k is algebraically closed, (2) k is an finite degree extension over L, (3) k can be given the structure of a normed vector space over L, (4) there is a natural projection from K to L. However, I hardly think that this could exhaust all cases. Any input would be helpful.