What is the ramification index of the extension field containing both $\alpha$ and $\beta$?

Question

Suppose $f(x)$ is an irreducible polynomial over the $p$-adic field $\mathbb{Q}_p$.

Let $\alpha, \beta$ be two roots of $f(x)$ with $p$-adic additive valuations $v(\alpha)=\frac{m}{p^2}$ and $v(\beta)=\frac{n}{p^2}$, for some natural numbers $m,n$.

What is the ramification index of the extension field containing both $\alpha$ and $\beta$ ?

How do we know whether two elements lies in the same extension field ?

Thanks