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Math Help - factorization of x^5-4

  1. #1
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    factorization of x^5-4

    Hey guys.

    I have an exam tomorrow, and one of my practice problems asks me to find the splitting field (and its degree) of x^5-4 over \mathbb{Q}.

    Clearly u=4^{1/5} is a root, and

    x^5-4=(x-u)(x^4+ux^3+u^2x^2+u^3x+u^4).

    Beyond that, though, I don't know where to go. I might just be missing something deceptively simple. Anyway, some assistance would be much appreciated!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    x^5=4 over complex field C contains 1 real root and 4 complex roots.
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    x^5=4 over complex field C contains 1 real root and 4 complex roots.
    Thank you, but what are those roots? That's what's giving me trouble.
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  4. #4
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    Quote Originally Posted by hatsoff View Post
    Thank you, but what are those roots? That's what's giving me trouble.

    The roots are 4^{1/5}, w 4^{1/5}, w^24^{1/5}, w^34^{1/5}, w^44^{1/5}, with w= a primitive root of 1 of order 5.

    Tonio
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