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Math Help - show that its an irreducible polynomial

  1. #1
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    show that its an irreducible polynomial

    Hello

    Consider the two finite fields \mathbb{F}_m and \mathbb{F}_n and let n be a prime number, q be a generator of \mathbb{F}_n^*.

    Now show that the polynomial \sum_{k=0}^{n-1} X^k is irreducible over \mathbb{F}_m -->


    Well if I define f(X):=\sum_{k=0}^{n-1} X^k then I think I'll have to assume that f(X)=h(X)*g(X) over \mathbb{F}_m with degree(h)>0 and degree(g)>0 ? But how does one do that?

    q has order n-1. What does that mean? Doesn't that mean that f(q) has order n-1 as well? How can I prove this?
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  2. #2
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    Re: show that its an irreducible polynomial

    You're missing some information on m. For example if K is the splitting field for f then K = \mathbb{F}_{n^t} as it is a field extension of  \mathbb{F}_n. So I think the theorem will hold if you maintain that  (m,n)=1.
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