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Math Help - Algebraic Closure.

  1. #1
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    Algebraic Closure.

    Does there exists a finite field such is algebraically closed?

    I tried of course the most obvious example \mathbb{Z}_2 but it is not algebraically closed for x^2+x+1\in F[\mathbb{Z}_2] has no zero.

    This problem seems strongly connected with number theory because it invloves prime numbers because all finite fields have order p^n.
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  2. #2
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    No finite field is algebraically closed. Consider the field of q elements (q some prime power). Then every field element satisfies the equation X^q = X, and so the polynomial X^q - X - 1 has no roots in the field.
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  3. #3
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    I was thinking about this today and I have a different approach.
    (For simplicity sake I am working with \mathbb{Z}_p only)

    1)Every finite field has form \mathbb{Z}_{p}.
    2)Assume that \mathbb{Z}_{p} is closed.
    3)Then x^2-a\in \mathbb{Z}_{p}[x],\forall a\in\mathbb{Z}_{p}
    4)Thus, x^2-a\equiv 0 \mod (p).
    5)Thus, x^2\equiv a \mod (p).
    6)Thus, (a/p)=1 for all a\in\mathbb{Z}_p, a\not =0 (Legendre symbol).
    7)Thus, \sum^{p-1}_{k=1}(k/p)=p-1.
    8)But, \sum^{p-1}_{k=1}(k/p)=0 (a theorem)
    9)A contradiction.
    10)Thus, no finite field of form \mathbb{Z}_{p} is algebraically closed.

    Now perhaps it can be completed for all field of  \mathbb{Z}_{p^n}
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