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Math Help - Is any algebraically closed field infinite ?

  1. #1
    Aki
    Aki is offline
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    Is any algebraically closed field infinite ?

    Is the following statement true ?
    If so, please explain why.

    If K is an algebraically closed field, then K is an infinite set.
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  2. #2
    Senior Member roninpro's Avatar
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    We know that all finite fields are extensions of fields \mathbb{Z}_p. And in particular, any finite extension of them will not be algebraically closed. So the algebraic closure of \mathbb{Z}_p is necessarily infinite.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by roninpro View Post
    We know that all finite fields are extensions of fields \mathbb{Z}_p. And in particular, any finite extension of them will not be algebraically closed. So the algebraic closure of \mathbb{Z}_p is necessarily infinite.
    Every finite field has order p^n for some n. The underlying multiplicative group is always cyclic (of order p^{n}-1). Thus, g^{p^n-1}=1 for all g \in \mathbb{F}_{n}\setminus \{0\}.

    Therefore, if p \neq 2, the equation x^{p^n-1}+2 will not have a root in \mathbb{F}_n.

    If p=2 then this trick doesn't work...however, if F=\{a_1, a_2, \ldots, a_{m}\} is your finite field, then the polynomial (x-a_1)(x-a_2)\ldots (x-a_{m})+1 has no roots in F (this works for all p).
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