This question is a bit vague, but I was wondering if someone might have an insightful answer.
Let $f_1$ and $f_2$ be irreducible polynomials in $\mathbb{Q}[x]$. Is there an easy criterion for knowing when the splitting fields of $f_1$ and $f_2$ yield the same field extensions of $\mathbb{Q}$?
Here is a related question. Let $L/\mathbb{Q}$ be a finite field extension. Assume both $f_1$ and $f_2$ remain irreducible in $L$. Given such an $L$, is there a way to determine when the splitting fields of $f_1$ and $f_2$ over $L$ are the same? (It is possible that the splitting fields of $f_1$ and $f_2$ over $\mathbb{Q}$ are different, but their splitting fields over $L$ are the same.)