Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$
When is it the case that $K=\mathbb Q(\alpha_i)$ for any $i$, i.e. when is the dimension of $K$ over $\mathbb Q$ equal to $n$, the degree of $f$? If this were true,then we would be able to express all the roots by rational combinations of each other. Is there a criterion that would decide whether an extension $\mathbb Q(\alpha)$ of $\mathbb Q$ has this property? Please explain your answers.