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Math Help - Polynomial Roots

  1. #1
    Senior Member vincisonfire's Avatar
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    Polynomial Roots

    Consider the case where  F = <br />
Z_p and the polynomial  X^{p^k} - X
    splits over K containing F . Show that the set
     GF_{p^k} = <br />
\{c \in K : c^{p^k} <br />
= c \} has exactly  p^k elements.
    I tried showing  X^{p^k} - X
    has  p^k distinct roots but did not succeed.
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  2. #2
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    Hi

    Since D(X^{p^k} - X)=-1,\ X^{p^k} - X has only distinct roots
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  3. #3
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    Here is another way to prove this that does not use derivatives.

    Let F be a field with charachteristic p then the polynomial x^n - 1 has n roots in a splitting field if p\not |n.

    We can write, if a is a zero then we can write x^n - 1 = (x-a)(x^{n-1}+x^{n-2}a+...+xa^{n-2}+a^{n-1}). But the second polynomial, x^{n-1}+...+a^{n-1}, when evaluated at a is a^{n-1}+a^{n-1}+...+a^{n-1} = n\cdot a^{n-1} \not = 0 since p\not | n.
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