abstract algebra - If a field extension of prime degree $p$ contains two roots of an irreducible polynomial then it contains all the roots.

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree p, with distinct roots $\alpha_1, \ldots, \alpha_p$. I want to show that if $F(\alpha_1) = F(\alpha_1, \alpha_2)$, then $F(\alpha_1) = F(\alpha_1, \ldots, \alpha_p)$, ie. $F(\alpha_1)$ is the splitting field of $f(x)$.

So far, what I know is that $F(\alpha_1) \simeq F(\alpha_i)$ for all $1 \leq i \leq p$, can I somehow extend this to show that $(F(\alpha_1))(\alpha_1) \simeq (F(\alpha_1))(\alpha_i)$? Thanks