If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:

$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$

The last non-trivial map here is "sum", with each local $Br(F_v)$ canonically injecting into $\mathbf{Q}/\mathbf{Z}$ by local class field theory.

In particular I can build a class of $Br(F)$ by writing down a finite number of elements $c_v\in Br(F_v)\subseteq \mathbf{Q}/\mathbf{Z}$, one for each element of a finite set $S$ of places of $v$, rigging it so that the sum $\sum_vc_v$ is zero in $\mathbf{Q}/\mathbf{Z}$.

This element of the global Brauer group gives rise to an equivalence class of central simple algebras over $F$, and if my understanding is correct this equivalence class will contain precisely one division algebra $D$ (and all the other elements of the equiv class will be $M_n(D)$ for $n=1,2,3,\ldots$).

My naive question: is the dimension of $D$ equal to $m^2$, with $m$ the gcd of the denominators of the $c_v$? I just realised that I've always assumed that this was the case, and I'd also always assumed in the local case that the dimension of the division algebra $D_v$ associated to $c_v$ was the square of the denominator of $c_v$. But it's only now, in writing notes on this stuff, that I realise I have no reference for it. Is it true??

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:

$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$

The last non-trivial map here is "sum", with each local $Br(F_v)$ canonically injecting into $\mathbf{Q}/\mathbf{Z}$ by local class field theory.

In particular I can build a class of $Br(F)$ by writing down a finite number of elements $c_v\in Br(F_v)\subseteq \mathbf{Q}/\mathbf{Z}$, one for each element of a finite set $S$ of places of $v$, rigging it so that the sum $\sum_vc_v$ is zero in $\mathbf{Q}/\mathbf{Z}$.

This element of the global Brauer group gives rise to an equivalence class of central simple algebras over $F$, and if my understanding is correct this equivalence class will contain precisely one division algebra $D$ (and all the other elements of the equiv class will be $M_n(D)$ for $n=1,2,3,\ldots$).

My naive question: is the dimension of $D$ equal to $m^2$, with $m$ the lcm of the denominators of the $c_v$? I just realised that I've always assumed that this was the case, and I'd also always assumed in the local case that the dimension of the division algebra $D_v$ associated to $c_v$ was the square of the denominator of $c_v$. But it's only now, in writing notes on this stuff, that I realise I have no reference for it. Is it true??