= Solution
For a complete discretely valued field with perfect <residue field> $k$, the <Witt residue sequence> is the split exact sequence
$$
0\longrightarrow\operatorname{Br}(k)\longrightarrow\operatorname{Br}(K)
\xrightarrow{\partial_K}H^1(k,\mathbb Q/\mathbb Z)\longrightarrow0.
$$
The first map identifies the unramified Brauer classes. The residue map is canonical; a splitting is chosen by a <uniformizer> $\pi$ and sends an unramified character $\chi$ to the cyclic Brauer class $(\chi,\pi)$. Changing $\pi$ may change this splitting by an unramified class. This is the version of Witt's theorem used here.
Now suppose $k=\mathbb F_q$. Fix arithmetic Frobenius $F_k:a\mapsto a^q$. Define the <local Brauer invariant> by
$$
\boxed{\operatorname{inv}_K(\alpha)=\partial_K(\alpha)(F_k)\in\mathbb Q/\mathbb Z}.
$$
Here is an explicit cohomological construction. Set $E=K^{\mathrm{nr}}$ and $G=\operatorname{Gal}(E/K)\cong\widehat{\mathbb Z}$. The permitted vanishing $\operatorname{Br}(E)=0$, together with <Hilbert 90> and inflation, identifies $\operatorname{Br}(K)$ with $H^2(G,E^\times)$. Normalize the valuation $v(\pi)=1$. Its induced map sends a multiplicative two-cocycle $c$ to the integer-valued two-cocycle $v(c)$, giving
$$
\operatorname{Br}(K)\longrightarrow H^2(G,\mathbb Z)
\xrightarrow{\ \delta^{-1}\ }H^1(G,\mathbb Q/\mathbb Z)
\xrightarrow{\ \chi\mapsto\chi(F_k)\ }\mathbb Q/\mathbb Z.
$$
The middle isomorphism is the connecting map for $0\to\mathbb Z\to\mathbb Q\to\mathbb Q/\mathbb Z\to0$, since positive-degree continuous cohomology of the trivial module $\mathbb Q$ vanishes.
To see that the valuation map is an isomorphism, its module sequence splits using $\pi$, and the unit part has zero second cohomology. For each finite unramified cyclic extension $E_m/K$, cyclic cohomology identifies that unit contribution with $\mathcal O_K^\times/N_{E_m/K}\mathcal O_{E_m}^\times$. The norm on residue-field units is surjective. On the principal-unit quotient of level $r\ge1$, the norm is the residue trace, through $N(1+\pi^rb)\equiv1+\pi^r\operatorname{Tr}(\overline b)\pmod{\pi^{r+1}}$. The trace of a finite separable <field extension> is surjective; successive corrections and completeness therefore make the norm surjective on all units. Passing to the direct limit gives the asserted vanishing. Finally, a continuous character of $\widehat{\mathbb Z}$ is uniquely specified by its value on arithmetic Frobenius, and that value can be any element of $\mathbb Q/\mathbb Z$. \b[The invariant map is an isomorphism.]
In concrete terms, if $E_m/K$ is the <unramified extension> of degree $m$ and $\sigma$ lifts arithmetic Frobenius, the <cyclic algebra> with $ub=\sigma(b)u$ and $u^m=a$ has
$$
\operatorname{inv}_K(E_m/K,\sigma,a)=\frac{v_K(a)}m\pmod{\mathbb Z}.
$$
Indeed its standard two-cocycle has value $a$ when the powers of $\sigma$ wrap past $m$ and value one otherwise. Valuation turns this into the connecting cocycle for the character with Frobenius value $v_K(a)/m$. This also fixes the sign of the normalization.
For <restriction and corestriction of local Brauer invariants>, let $l$ be the <residue field> of $L$, $e$ the <ramification index> and $f=[l:k]$. A finite extension of <local fields> has $[L:K]=ef$, $v_L|_K=e v_K$ and $v_K(N_{L/K}b)=f v_L(b)$. Every class of $\operatorname{Br}(K)$ is $(\chi,\pi_K)$, with $\chi$ unramified. After restriction to $L$, its character has Frobenius value $f\chi(F_k)$ and its parameter has valuation $e$. Thus
$$
\boxed{\operatorname{inv}_L(\operatorname{Res}_{L/K}\alpha)=[L:K]\operatorname{inv}_K(\alpha)}.
$$
For corestriction, write a class of $\operatorname{Br}(L)$ as $(\chi_l,\pi_L)$. Choose a character $\chi_k$ of the finite-field absolute group such that $\operatorname{Res}_{l/k}\chi_k=\chi_l$. This is possible because restriction is multiplication by $f$ on $\mathbb Q/\mathbb Z$, which is divisible. Inflate both characters to the <local fields>. The <norm projection formula for cyclic Brauer pairings> gives
$$
\operatorname{Cor}_{L/K}(\chi_l,\pi_L)=(\chi_k,N_{L/K}\pi_L).
$$
This projection identity follows from the cohomological transfer and cup-product identity; on multiplicative degree-zero coefficients transfer is the <field norm>, and the identity in higher degree follows by the cochain definition or dimension shifting. The norm parameter has $K$-valuation $f$, so its invariant is $f\chi_k(F_k)=\chi_l(F_l)$. Therefore
$$
\boxed{\operatorname{inv}_K(\operatorname{Cor}_{L/K}\beta)=\operatorname{inv}_L(\beta)}.
$$
This argument uses the character-and-norm description directly and does not cancel multiplication by $[L:K]$ in $\mathbb Q/\mathbb Z$, where such cancellation would be invalid. For purely inseparable steps in equal characteristic, <absolute Galois groups> identify, restriction uses inclusion of multiplicative coefficient modules, and Brauer corestriction uses the <field norm> on them. The same projection calculation applies. Factoring a general finite extension into separable and purely inseparable steps covers all finite extensions in the question.
Lastly, a class with invariant $a/b$ in lowest terms has <Brauer period> $b$. The <unramified extension> of degree $b$ splits it, so its division-algebra degree divides $b$. Conversely a maximal subfield of the <division algebra> splits it and has degree equal to that division-algebra degree; the restriction formula forces $b$ to divide that degree. These standard splitting-field properties are among the permitted inputs. Thus the <index of a central simple algebra> equals its period over a <local field>. The <central division algebras> of degree $n$ are exactly those with invariants $a/n$ and $\gcd(a,n)=1$. The Brauer class determines its division representative uniquely, so there are
$$
\boxed{\varphi(n)}
$$
isomorphism classes, where $\varphi$ is the Euler totient function. This includes the single degree-one algebra $K$.
Back to article page
