Brauer Group of $\mathbb{Q}_2$

I have read that the Brauer Group of any local field is $\mathbb{Q}/\mathbb{Z}$, and I want to see this for $\mathbb{Q}_2$. I'm confused because it appears there are $3$ division algebras of dimension $4$, namely $\mathbb{Q}_2(u, v)/(u^2=a, v^2=b, uv=-vu)$ for $a, b$ are any pair among $2, 3, 5$. What am I missing?

Solution 1:

I discovered what is wrong. If $a=2, b=3$, and we let $x\in\mathbb{Q}_2$ be a solution to $x^2=17$, which exists by Hensel, then $\left(\frac{xv+uv}{3}\right)^2=5$, by anticommutativity, and $\frac{xv+uv}{3}$ and $u$ anticommute, so the three division algebras are actually all the same. So there's only one division algebra as expected.

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