 Local Brauer invariant

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.