Albert-Brauer-Hasse-Noether theorem ($0\to \mathrm{Br}(K)\to {⨁}_v\mathrm{Br}(K_v)\stackrel{\sum {\mathrm{inv}}_v}{\to }\mathbb{Q}/\mathbb{Z}\to 0$)

For a number field $K$, localization of the Brauer group is injective. Its image is exactly the finite-support tuples of local classes whose local Brauer invariants sum to zero. The non-Archimedean summands have invariant group $\mathbb{Q}/\mathbb{Z}$, real summands have $\{0,1/2\}$, and complex summands are zero. For a cyclic character $\chi$ and parameter $a$, the invariant of the local cyclic algebra is $\chi ({\mathrm{Art}}_{K_v}(a))$. Thus the zero-sum condition on principal parameters is the additive form of Artin reciprocity.