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Math Help - Two algebra questions

  1. #1
    Junior Member
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    Two algebra questions

    I have problem with these two exercise:

    1. Assume, that \epsilon \in C is a nth root of unity. Show: if \epsilon \neq 1, then \epsilon is the root of the syclotomic polynomial \Phi_n = 1 + X + X^2 + \dots + X^{n-1} \in \mathbb{C}[X].
    ( \epsilon = e^{2\pi i / n})

    2. Show: if k \in \mathbb{C}, then X^n - k^n = (X-k)(X-\epsilon k) \cdots (X-\epsilon^{n-1}k), where \epsilon = e^{2\pi i / n}.

    I have no idea how to begin with these, so any help would be nice.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hints:

    1. X^n-1=(X-1)\Phi_n(X) .

    2. Clearly, k is a root of X^n-k^n . Prove that \epsilon^jk is also a root for every j=1,\ldots,n-1


    Fernando Revilla
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Further to FernandoRevilla's reply, basically an n-th root of unity is a corner of a regular n-gon inscribed in the uni circle.

    For example,



    This is because it must satisfy the equation x^n-1=0. That is where the \epsilon=e^{2 \pi i/n} comes from.
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