# Is there an explicit description of maximal extension of $\mathbb{Q}$ unramified at a particular prime?

If we write $\mathbb{Q}^{un,p}$ for the maximal algebraic extension of $\mathbb{Q}$ which is unramified at $p$, is there a nice description of $\mathbb{Q}^{un,p}$? I have a suspicion the answer to this is that we have no idea, but perhaps not.