  Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect
closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring
$A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that
the ring $A\otimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal
power series in $n$ indeterminates over $k$.
