Algorithm for determining whether two polynomials have the same splitting field

This question 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.)

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This question just boils down to the question of whether a polynomial with coefficients in a given number field has a root (apply this repeatedly), and there are algorithms for this -- see e.g. Cohen's "a course in number theory and cryptography". –  user30035 Dec 31 '12 at 8:52