  Let $F$ be a non archimedean local field, and $n_1$ and $n_2$ two positive
even integers. We prove that if $\pi_1$ and $\pi_2$ are two smooth
representations of $GL(n_1,F)$ and $GL(n_2,F)$ respectively, both admitting a
Shalika period, then the normalised parabolically induced representation
$\pi_1\times \pi_2$ also admits a Shalika period. Combining this with the
results of \cite{M-localBF}, we obtain as a corollary the classification of
generic representations of $GL(n,F)$ admitting a Shalika period when $F$ has
characteristic zero. This result is relevant to the study of the
Jacquet-Shalika exterior square $L$ factor.
