Let $p, q \in \mathbb{Q}[T]$ be irreducible and with degree greater than one (and WLOG monic). When is it that $\mathbb{Q}[T]/(p(T)) \cong \mathbb{Q}[T]/(q(T))$?
Take any roots $\alpha, \beta$ of $p, q$ (resp). We then have that the condition on the question is equivalent to how to decide whether $\mathbb{Q}(\alpha) \cong \mathbb{Q}(\beta)$ is true or not.
Going further, is there a simple way to characterize all the primitive elements of finite simple extensions $L = K(\alpha)$ where $[L:K] = d > 1$?
If we have $K = \mathbb{Q}$ and $L = \mathbb{Q}(\sqrt{2}, \sqrt{3})$, we have $L = K(\sqrt{2}+\sqrt{3}) = K(\sqrt{2} - \sqrt{3}) = K(\sqrt{2}+\sqrt{6})$.
In fact, I'm pretty sure that a lot (almost all) of linear combinations of $1, \sqrt{2}, \sqrt{3}, \sqrt{6}$ work. Is there a method for determining precisely which of them work? Intuitively it seems to suffice to me that two of these roots appear on the linear combination. How would that generalize to, say, $\mathbb{Q}(\sqrt{2}+\sqrt{3}+\sqrt{5})$?
I am aware that a proof for the primitive element theorem builds a primitive element. Maybe we can we adapt it to characterize all of them?