Hilbertʹs Fourteenth Problem and algebraic extensions

Let k[X] be the polynomial ring in n variables over a field k for some n N, and k(X) its field of fractions. Assume that L is a subfield of k(X) containing k over which k(X) is algebraic. In spite of being an important issue in Hilbertʹs Fourteenth Problem, relations between finite generation of the k-subalgebra L∩k[X] of k[X] and the extension degree [k(X):L] of k(X) over L have not been investigated. In the present paper, we give an example of L with [k(X):L]=d such that L∩k[X] is not finitely generated for each d N with d 3 for n=3.