*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?