As an example, consider the polynomial $f(x)=x^2-2$. This polynomial has no roots over the rationals $\mathbb{Q}$. However, $\sqrt{2} \in \mathbb{Q}_7$, so it does have roots in the 7-adics. Is there a name for this property? Is there a more elegant way of saying: Suppose $f(x)$ is an irreducible polynomial over $\mathbb{Q}$ and $p$ is prime for which $f(x)$ has all its roots in $\mathbb{Q}_p$.
Or, alternatively, if you have an algebraic $\alpha$, is there an established terminology the describes whether or not $\mathbb{Q}_p$ contains an isomorphic copy of the extension $\mathbb{Q}(\alpha)$?