Local fundamental class ($u_{L/K}\in H^2(\mathrm{Gal}(L/K),L^{\times }),{\mathrm{inv}}_K(u_{L/K})=1/[L:K]$)

For a finite Galois extension $L/K$ of non-Archimedean local fields, its relative Brauer group is the kernel of restriction $\mathrm{Br}(K)\to \mathrm{Br}(L)$ and identifies with $H^2(\mathrm{Gal}(L/K),L^{\times })$. Restriction multiplies the local Brauer invariant by $[L:K]$. This kernel is therefore cyclic of order $[L:K]$, and the indicated class is its distinguished generator. In an unramified extension with arithmetic Frobenius generator it is represented by the cyclic algebra with parameter a uniformizer.