= Albert-Brauer-Hasse-Noether theorem
{c}
{title2=$0\to\operatorname{Br}(K)\to\bigoplus_v\operatorname{Br}(K_v)\xrightarrow{\sum\operatorname{inv}_v}\mathbb Q/\mathbb Z\to0$}
= Global Brauer invariant sequence
{synonym}
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 $\chi(\operatorname{Art}_{K_v}(a))$. Thus the zero-sum condition on principal parameters is the additive form of <Artin reciprocity>.
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