Let $E$ be a separable quadratic algebra over a locally compact field $F$ of
positive characteristic. The Langlands-Shahidi method can be used to define the
Asai $\gamma$-factors for a smooth irreducible generic representation $\pi$ of
$\GL_n(E)$. If $\sigma$ is the Weil-Deligne representation of $\mathcal{W}_E$
corresponding to $\pi$ under the local Langlands correspondence, then it is
shown that the Asai $\gamma$-factor is the same as the $\gamma$-factor on the
Galois side corresponding to the representation of $\mathcal{W}_E$ obtained
from $\sigma$ under tensor induction.