A finite field $\mathbb{F}_p$ posesses the property that for any irreducible polynomial $f\in\mathbb{F}_p[x]$ adjoining any root of $f$ automatically adjoins all roots of $f$. (In other words, any extension of $\mathbb{F}_p$ is normal.)
Does the field $\mathbb{Q}_p$ posess the same property? If no, is it true for irreducible polynomials $f\in\mathbb{Q}_p[x]$ with coefficients in the set $\{0,1,...,p-1\}$?
The question is whether a finite extension $L$ of our given field $K$ is normal, or not. If it is normal, then every irreducible polynomial in $K[x]$ that has one root in $L$, has all of its roots in $L$.
Finite extensions of finite fields are Galois, hence normal. Finite extensions of $\mathbb{Q}_p$, however, need not be normal in general. However, if the extension $L/K$ is unramified then $L/K$ is Galois (see here).