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Math Help - reducible polynomial x^p+a in Zp

  1. #1
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    reducible polynomial x^p+a in Zp

    Let p be a prime and a be an element of Zp. Show that x^p+a is reducible in Zp[x]

    im not sure if this is the best way to approach the question but
    as x^p+a is reducible there must be a factor
    therefore x^p+a=0
    and x^p=-a for some x

    would it then suffice to say that as Zp contains a primitive root,
    hence there exists an element of Zp congruent to -a, and therefore there is a root in Zp? This doesn't seem like a very good solid solution
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  2. #2
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    Quote Originally Posted by i_never_noticed View Post
    Let p be a prime and a be an element of Zp. Show that x^p+a is reducible in Zp[x]

    im not sure if this is the best way to approach the question but
    as x^p+a is reducible there must be a factor
    therefore x^p+a=0
    and x^p=-a for some x

    would it then suffice to say that as Zp contains a primitive root,
    hence there exists an element of Zp congruent to -a, and therefore there is a root in Zp? This doesn't seem like a very good solid solution

    It really isn't. Try to show the following:

    === The map a\to a^p in \mathbb{Z}/p\mathbb{Z} is a bijection (in particular it is onto) ===

    Now solve your problem

    Tonio
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