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Math Help - Splitting field of an irreducible polynomial

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    MHF Contributor Bruno J.'s Avatar
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    Splitting field of an irreducible polynomial

    Suppose K is the splitting field of an irreducible polynomial p(x) \in \mathbb{Z}[x]. What is a general condition to have [K:\mathbb{Q}] = \mbox{ deg }p(x)? Is there a systematic way to test if that is the case, given p(x)?
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    Quote Originally Posted by Bruno J. View Post
    Suppose K is the splitting field of an irreducible polynomial p(x) \in \mathbb{Z}[x]. What is a general condition to have [K:\mathbb{Q}] = \mbox{ deg }p(x)? Is there a systematic way to test if that is the case, given p(x)?
    as far as i know there's no general condition in this case but if we replace \mathbb{Q} with any finite field, then the result is always true. (the proof is quite easy!)
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