  Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and
$k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's
fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$
is finitely generated. Various counterexamples to this problem were already
given, but the case $[k({\bf x}):L]=2$ was open when $n=3$. In this paper, we
study the problem in terms of the field-theoretic properties of $L$. We say
that $L$ is minimal if the transcendence degree $r$ of $L$ over $k$ is equal to
that of $A$. We show that, if $r\ge 2$ and $L$ is minimal, then there exists
$\sigma \in {\mathop{\rm Aut}\nolimits}_kk(x_1,\ldots ,x_{n+1})$ for which
$\sigma (L(x_{n+1}))$ is minimal and a counterexample to the problem. Our
result implies the existence of interesting new counterexamples including one
with $n=3$ and $[k({\bf x}):L]=2$.
