[This question](https://math.stackexchange.com/questions/34522/cubic-polynomials-that-generate-the-same-extension) asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. Also, it's not clear how the results generalize to higher degree polynomials. Is there an algorithm for determining whether two polynomials in in $\mathbb{Q}[x]$ have the same splitting field? If so, what is it, and why does it work?


(Reposted [from Math Stack Exchange](https://math.stackexchange.com/questions/267873/algorithm-for-determining-whether-two-polynomials-have-the-same-splitting-field).)



Algorithm for determining whether two polynomials have the same splitting field
In practice, you can reduce f and g mod the first 1000 primes (deleting those where f and g are ramified) and see if the primes all have the same splitting behavior. If they don't, the splitting fields are not the same, and if they do, you can be pretty darn sure that they are. If you want to replace "be pretty darn sure" with "have proved" you can use the effective Chebotarev of Lagarias and Odlyzko to figure out the value of 1000 which actually gives you a proof.


