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I've been reading the section about Brauer groups in Introduction to Modern Number Theory, and I couldn't quite understand how this isomorphism is defined.

We start with a central simple algebra $A$, over $\mathbb{Q}_p$. Then, the book says, we take ``a maximal unramified extension'' $L$ of $\mathbb{Q}_p$ in $A$. The book informs me that there are several of those, and that they are all conjugate in $A$. Then the book says the the valuation on $\mathbb{Q}_p$ extends to a valuation $v_A$ on $A$, and that the Frobenius automorphism of $L$ over $\mathbb{Q}_p$ extends to an inner automorphism of $A$ by some element $\gamma \in A$. Thus $v_A:A^{\times}\rightarrow \frac{1}{n}\mathbb{Z}$, and we define the map that is designed to be an isomorphism from $Br(\mathbb{Q}_p)$ to $\mathbb{Q}/\mathbb{Z}$ by $A\mapsto v_A(\gamma)$. The book informs me that this is well defined and in fact an isomophism.

But I'm not sure I understand what $L$ being a ``maximal unramified extension'' means. Is it: let $L$ be a field extension of $\mathbb{Q}_p$, contained in $A$, and unramified over $K$? I can't think of any examples where $L$ wouldn't be $K$ itself, since the center of $A$ is $\mathbb{Q}_p$? Is this because my intuition is bad (if so please give me an example), or is it because I have the wrong definition in mind?

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    Your confusion over how there could be multiple copies of $L$ inside $A$ can be removed by looking at an actual example. Take $A = {\mathbf H}({\mathbf Q}_2)$, the quaternions over the 2-adics. Convince yourself this is a division ring. There are a *lot* of solutions in $A$ of $q^2 = -3$ (one example is $q = i+j+k$) and for any of them, ${\mathbf Q}_2[q]$ is a copy of ${\mathbf Q}_2(\sqrt{-3})$ inside $A$, and this is a quad. unram. extn. of ${\mathbf Q}_2$. Do you know there are infinitely many copies of ${\mathbf C}$ in ${\mathbf H}({\mathbf R})$?2012-04-28

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