Let \(k\) be a field of characteristic \(0,\) \(k[X]=k[X_{1},....,X_{n}]\) the polynomial ring in \(n\) variables over \(k\) and \(k(X)\) the field of fractions of \(k[X].\) In this paper, in response to Hilbert's fourteenth problem, for \(n=3\) and each \(d\geq 3\) constructive examples have been given to show that there exists a subfield \(L\) of \(k(X)\) containing \(k\) such that \([k(X):L]=d\) and \(L\cap k[X]\) is not a finitely generated \(k\)-algebra. Such counter examples are already known for large \(n.\) Some related open problems are also stated.