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?


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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    $\begingroup$ Good job figuring this out! But the real reason I comment here is about those three AwesomeMath entrance questions you flagged. There is little we can do about the ones that received a solution - the horse bolted. Are you in touch with the organizers? Just informing them about this is possibly the best course of action now. We are not allowed to disclose personally identifying information, but I guess I can tell that the system tracked that user to a continent quite distant from North America. Temporarily deleting those questions will prevent others from seeing them. Doing that $\endgroup$ Jan 12 '17 at 5:05

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