On the Brauer Group of a Two-Dimensional Local Field

The two-dimensional local field $K=F_q((u))((t))$, $\operatorname{char}K=p$, and its Brauer group $\operatorname{Br}(K)$ are considered. It is proved that, if $L=K(x)$ is the field extension for which we have $x^p-x=ut^{-p}=:h$, then the condition that $(y,f\,|\,h]_K=0$ for any $y\in K$ is equivalent to the condition $f\in\operatorname{Im}(\operatorname{Nm}(L^*))$.