= Solution
A <central simple algebra> over $K$ is a finite-dimensional associative $K$-<algebra> with no nonzero proper two-sided <ideals> and with center $K$. The <Artin–Wedderburn theorem> writes it uniquely as $M_r(D)$ with $D$ a central <division algebra>. After a finite separable <splitting field> extension it becomes $M_d$, so its dimension is $d^2$, and $d$ is its <degree of a central simple algebra>. Passing from $D$ to a <matrix algebra> over $D$ does not change its <Brauer class>.
The <Brauer group> consists of these classes, with product induced by <tensor product>. Extending to a common <splitting field> shows that a <tensor product of central simple algebras> is again central simple; its two-sided <ideals> and center descend from the corresponding full <matrix algebra>. The neutral class is $[K]$, and the inverse of $[A]$ is $[A^{\mathrm{op}}]$, because
$$
A\otimes_K A^{\mathrm{op}}\cong\operatorname{End}_K(A),\qquad a\otimes b\mapsto(x\mapsto axb).
$$
The displayed map becomes the standard isomorphism for matrices over a <splitting field>, so it is already an isomorphism over $K$. Thus <Brauer classes> measure precisely the obstruction to being a split <matrix algebra>.
An explicit family is supplied by <cyclic algebras>. Let $L/K$ be cyclic of degree $n$, with generator $\sigma$, and let $a\in K^\times$. Define
$$
A=(L/K,\sigma,a)=\bigoplus_{i=0}^{n-1}Lu^i,\qquad ux=\sigma(x)u,\qquad u^n=a.
$$
This is an associative <algebra>: form the skew polynomial ring with relation $ux=\sigma(x)u$ and quotient by the central relation $u^n-a$. Reduction of powers gives the displayed $n^2$-dimensional $K$-<basis>. An element commuting with every $x\in L$ has no $u^i$ coefficient for $i\ne0$, since $\sigma^i$ is not the identity. Commuting also with $u$ forces the remaining coefficient into $K$. Hence the center is $K$.
Here is an explicit splitting construction. On $L^n$, with <basis> $e_0,\ldots,e_{n-1}$, put
$$
D(x)e_j=\sigma^j(x)e_j,\qquad Ue_j=e_{j-1}\ (j\ge1),\qquad Ue_0=a e_{n-1}.
$$
Then $UD(x)=D(\sigma x)U$ and $U^n=aI$. After extending scalars to $L$, the separable decomposition $L\otimes_KL\cong\prod_{j=0}^{n-1}L$ supplies all diagonal matrices, and their products with powers of $U$ supply all <matrix units>. Therefore
$$
A\otimes_KL\cong M_n(L).
$$
Any nonzero proper two-sided <ideal> of $A$ would extend to one of $M_n(L)$, which is impossible. Thus this construction really gives a <central simple algebra>, with <splitting field> $L$.
Its central arithmetic feature is the <splitting criterion for a cyclic algebra>:
$$
\boxed{(L/K,\sigma,a)\text{ splits}\iff a\in N_{L/K}L^\times.}
$$
If $a=Nc$, let $L$ act by multiplication on the $K$-<vector space> $L$, and let $u$ act as $c\sigma$. Its nth power is multiplication by $Nc=a$, so this gives a representation $A\to\operatorname{End}_K(L)$. It is injective by simplicity and an isomorphism by equal dimension. Conversely, if $A\cong\operatorname{End}_K(V)$ with $\dim_K V=n$, its embedded field $L$ makes $V$ one-dimensional over $L$. The relation for $u$ says that $u$ is $\sigma$-semilinear. In an $L$-<basis> it must have the form $c\sigma$, whose nth power is $Nc$. The relation $u^n=a$ forces $a=Nc$.
More generally the <crossed-product algebra of a Galois extension> attaches an <algebra> to a multiplicative two-cocycle. Changing the cocycle by a coboundary rescales its basis and preserves the <Brauer class>. Descent of a split <matrix algebra> identifies
$$
\ker\bigl(\operatorname{Br}(K)\to\operatorname{Br}(L)\bigr)\cong H^2(\operatorname{Gal}(L/K),L^\times).
$$
For cyclic $L/K$, cyclic <Tate cohomology> identifies this second <cohomology group> with $K^\times/NL^\times$. In a crossed product, rescale successive cyclic basis elements to powers of a single $u$; the remaining datum is $u^n=a$, changed by a <field norm> when $u$ is rescaled. This explains why <cyclic algebras> give every class split by a cyclic extension, and why multiplication of their parameters is addition of their <Brauer classes>.
For a <finite field> $K=\mathbb F_q$, every finite <splitting field> $L=\mathbb F_{q^n}$ is cyclic over $K$. The <field norm> is the power map $x\mapsto x^{(q^n-1)/(q-1)}$ on its cyclic multiplicative <group>, and is onto $\mathbb F_q^\times$. Thus every relative <Brauer class> is zero, and every <central simple algebra> has zero class:
$$
\boxed{\operatorname{Br}(\mathbb F_q)=0.}
$$
For a non-Archimedean <local field> $K$, the valuation classification of central <division algebras> shows that every <Brauer class> splits over a finite <unramified extension>. For the degree-$n$ <unramified extension> $L/K$ with arithmetic <Frobenius> $\sigma$, units are norms and norm <valuations> are multiples of $n$. Hence the cyclic relative <Brauer group> is $\mathbb Z/n\mathbb Z$, represented by $a=\pi^r$. The compatible <local Brauer invariant> is
$$
\operatorname{inv}_K[(L/K,\sigma,a)]=\frac{v_K(a)}n\pmod{\mathbb Z}.
$$
The union over all $n$ gives
$$
\boxed{\operatorname{Br}(K)\cong\mathbb Q/\mathbb Z.}
$$
A class with invariant $r/n$ in lowest terms has a central <division algebra> representative of <degree of a central simple algebra> $n$. This is the local valuation classification used above; the explicit <cyclic algebras> exhibit all of its invariants. At the Archimedean places, $\operatorname{Br}(\mathbb C)=0$, while $\operatorname{Br}(\mathbb R)=\mathbb Z/2\mathbb Z$: its nontrivial class is the <quaternion algebra> $(\mathbb C/\mathbb R,\text{conjugation},-1)$, since complex <norms> are positive.
For a <number field>, the <Albert-Brauer-Hasse-Noether theorem> describes the <Brauer group> by the exact sequence
$$
0\longrightarrow\operatorname{Br}(K)\longrightarrow\bigoplus_v\operatorname{Br}(K_v)\xrightarrow{\ \sum_v\operatorname{inv}_v\ }\mathbb Q/\mathbb Z\longrightarrow0.
$$
Thus its classes are exactly finite-support families of local invariants with sum zero; real invariants are $0$ or $1/2$ and complex invariants are zero. Every global <Brauer class> has a cyclic splitting extension, so <cyclic algebras> also realize all global classes. If $\chi(\sigma)=1/n$, their local invariants are $\chi_v(\operatorname{rec}_{K_v}(a))$, with $\chi_v$ restricted to the local decomposition subgroup. The sum-zero relation for a principal parameter is the invariant form of <Artin reciprocity>. Injectivity supplies the local-to-global splitting test, which yields the <Hasse norm theorem> for cyclic extensions as explained in the preceding solution. Here the local valuation classification and the global invariant theorem are the structural theorems used to describe the <Brauer groups>; the construction and norm-splitting calculation have been proved explicitly.
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