What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes

Question

A smart man once explained to me how to solve the following problem, then I forgot.

Let $F\subset\mathbb{R}$ be a number field, let $d\in F^+$, and let $K=F(\sqrt{-d})$. Denote the rings of integers of $F$ and $K$ respectively by $\mathbb{Z}_F$ and $\mathbb{Z}_K$. Suppose $\mathfrak{p}\vartriangleleft\mathbb{Z}_F$ is a prime ideal. Then there exists some prime ideal $\mathfrak{P}\vartriangleleft\mathbb{Z}_K$ such that one of the following things is true.

1. $\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}$
2. $\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}^2$
3. $\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}\overline{\mathfrak{P}}$

Now let $\mathcal{A}$ be a quaternion algebra over $F$ that is ramified at the place corresponding to $\mathfrak{p}$, and let $\mathcal{B}=\mathcal{A}\otimes_FK$ (the quaternion algebra over $K$ resulting from the extension of scalars).

Is $\mathcal{B}$ ramified at $\mathfrak{P}$ (and at $\overline{\mathfrak{P}}$ in case 3)? Why or why not?

Answer 1

The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K_{\mathfrak{P}}/F_{\mathfrak{p}}$ has odd degree, i.e. case 3. A general way to see this kind of property is that the invariant in $\mathbb{Q}/\mathbb{Z}$ of the class in the Brauer group gets multiplied by the degree when you base-change the corresponding algebra. A more down-to-earth proof in the case of quaternion algebra is to use the standard presentation of the division quaternion algebra over the local field $F_{\mathfrak{p}}$ and to see that it contains all the quadratic extensions of $F_{\mathfrak{p}}$; in case 3 the local extension is trivial and so locally there is no base-change at all.

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Edit: Here are some references. Purely on $p$-adic fields, I guess Serre's Local fields. On quaternion algebras, there is the classic Algèbre de quaternions by Vignéras, but it is in French; there is the forthcoming book on quaternion algebras by John Voight, which is looking to be amazing; given your taste for geometry, I would also recommend the excellent The arithmetic of hyperbolic three-manifolds by Maclachlan and Reid, that contains very nice chapters on the local structure and the arithmetic of quaternion algebras. On more general central simple algebras and Brauer groups, I like the section in Milne's Class Field Theory notes, but this material is also contained in many textbooks.