Question on a finite field extension of $\mathbb{Q}$

Question

I have a polynomial $p(x) \in \mathbb{Q}[x]$ and is irreducible over $\mathbb{Q}$. Let it be of degree $n$ and $\alpha_1, ..., \alpha_n$ be its roots. I know that $$ \mathbb{Q}(\alpha_i) \cong \mathbb{Q}(\alpha_j). $$ What I was curious is does $\mathbb{Q}(\alpha_1) = \mathbb{Q}(\alpha_1, ..., \alpha_n)$? If this is not the case, is there a known formula for $[\mathbb{Q}(\alpha_1, ..., \alpha_n) : \mathbb{Q} ]$? Thanks!