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.)

abelian(over $\mathbb{Q}$) there might be a way. Supoose so and both are monic with integer coefficients (too easy?) First, test if they have real or complex roots (by using calculus to determine whteher they have one, and hence all, real root). Now you know ramification at infinity. Secondly, look at the discriminants: they are the discriminants of the orders $\mathbb{Z}(\alpha_i), i=1,2$ and let $D$ be the l.c.m. Now you need to look at... – Filippo Alberto Edoardo Apr 3 '13 at 23:46