= Solution
The <Brauer group> of a <field> consists of <equivalence classes> of <central simple algebras>, with addition induced by <tensor product>, zero represented by <matrix algebras>, and inverse represented by the <opposite algebra>. Localization sends a global algebra to its scalar extensions at the completions. For a non-Archimedean completion of a <number field>, the <local Brauer invariant> is an <isomorphism>
$$
\operatorname{inv}_v:\operatorname{Br}(K_v)\xrightarrow{\sim}\mathbb Q/\mathbb Z.
$$
Normalize it so that an unramified <cyclic algebra> of degree $m$, with <arithmetic Frobenius> generator and parameter $a$, has invariant $v(a)/m$. For a real completion, $\operatorname{Br}(\mathbb R)=\mathbb Z/2$ with the Hamilton <quaternion algebra> mapped to $1/2$; for a complex completion the <Brauer group> is zero.
The global computation is the <exact sequence> in the <Albert-Brauer-Hasse-Noether theorem>:
$$
\boxed{0\longrightarrow\operatorname{Br}(K)
\longrightarrow\bigoplus_v\operatorname{Br}(K_v)
\xrightarrow{\sum_v\operatorname{inv}_v}\mathbb Q/\mathbb Z
\longrightarrow0.}
$$
Thus a global Brauer class is determined uniquely by a finite-support family of local invariants. Such a family occurs if and only if its sum is zero; at real places the only allowed entries are $0,1/2$, and at complex places the entry is zero. <Injectivity> is a local-global splitting principle; the middle exactness is the reciprocity constraint; <surjectivity> of the sum follows already from any one finite local summand. These are separate assertions, not merely different descriptions of the sum formula.
For an explicit <group> description, choose one finite place $v_0$. Assign arbitrary finite-support invariants at all other finite places and arbitrary allowed invariants at the real places, then set the invariant at $v_0$ equal to minus their sum. This gives
$$
\operatorname{Br}(K)\cong
\left(\bigoplus_{v\ {
m finite},\ v\ne v_0}\mathbb Q/\mathbb Z\right)
\oplus(\mathbb Z/2)^{r_1}.
$$
The description by the entire invariant family is canonical; this displayed elimination of one coordinate depends on the choice of $v_0$.
The connection with <Artin reciprocity> is especially transparent for <cyclic algebras>. Let $\chi:\operatorname{Gal}(L/K)\hookrightarrow\mathbb Q/\mathbb Z$ describe a cyclic extension, with $\chi(\sigma)=1/n$. Write $(\chi,a)$ for the class of $(L/K,\sigma,a)$. The local reciprocity pairing is
$$
\operatorname{inv}_v(\chi_v,a_v)
=\chi_v\bigl(\operatorname{Art}_{K_v}(a_v)\bigr).
$$
It follows from the <local fundamental class> construction of the <Local Artin map>; on an unramified character it is precisely the normalization $v(a_v)/n$. The <group kernel> in the parameter is the local norm <group>, and restriction/corestriction give the norm compatibility of this pairing.
For a principal parameter $a\in K^\times$, the zero-sum identity for its global <cyclic algebra> therefore reads
$$
\sum_v\chi_v(\operatorname{Art}_{K_v}(a))=0.
$$
On the other hand the idelic <Artin reciprocity law> is
$$
\prod_v\operatorname{Art}_{L/K,v}(a)=1
\quad(a\in K^\times),
$$
and evaluating this product by $\chi$ gives exactly that zero-sum identity. Conversely, testing every cyclic character of a finite abelian <Galois group> separates its elements, so the zero-sum identities for <cyclic algebras> imply this principal-idele reciprocity law for every finite <abelian extension>. At almost all places a unit and an unramified character pair trivially, so the sums and products are finite.
The full idelic statement identifies the <group kernel> as $K^\times N_{L/K}J_L$ and induces $J_K/(K^\times N_{L/K}J_L)\cong\operatorname{Gal}(L/K)$. It explains why the local norm-residue pairings must glue globally. The global Brauer theorem supplies, in addition, the <injectivity> and realization statements needed to compute all algebra classes; the principal-product identity alone would not prove those statements. For <cyclic algebras> its <injectivity> yields the <Hasse norm theorem>, as in Question 3. In degree two, an invariant is either zero or $1/2$, so the zero-sum law becomes the <Hilbert reciprocity law> $\prod_v(a,b)_v=1$. For example, a <quaternion algebra> over $\mathbb Q$ can be ramified at precisely a prescribed <finite set> of places, including possibly the real place, if and only if that set has even cardinality; the invariants uniquely determine its Brauer class. This illustrates how arithmetic reciprocity becomes a concrete constraint on global algebra structure.
Back to article page
