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


*

*$\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}$

*$\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}^2$

*$\mathfrak{p}\mathbb{Z}_K=\mathfrak{P}\overline{\mathfrak{P}}$


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?
 A: 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.

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.
