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We know that all finite fields are perfect (fields with char $p$). Also fields with char 0 (infinite fields) are perfect. Then what are the fields that are not perfect?

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    May I ask you to post an answer to your own question? In such a way you will be sure that your intuition is correct and the question will not remain in the "unanswered" category forever! If you don´t have time to do that just let it know to someone who can answer. Thank you and welcome to Math.Se!2012-02-07

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Example of non-perfect field: $\,\mathbb F_p(T)=\,$ the field of rational functions in an unknown (transcendental element) $\,T\,$ .

Why? The polynomial $\,f(x)=x^p-T\in\mathbb F_p(T)[x]\,$ is

$\,(1)\,\,$ irreducible: Apply Eisenstein's Criterion in the UFD $\,\mathbb F_p[T]\subset \mathbb F_p(T)\,$ and the prime $\,T\,$ in it

$\,(2)\,\,$ Let $\,\alpha\,$ be some root of $\,f(x)\,$ in some field extension, then $\alpha^p=T\Longrightarrow x^p-\alpha^p=(x-\alpha)^p\in\mathbb F_p[T]$and thus $\,\alpha\,$ is the unique root of $\,f(x)\,$, what makes this irreducible polynomial as inseparable as one could ever hope and, thus, the field $\,\mathbb F_p(T)\,$ is non-perfect.

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    interesting... I guess that makes sense, since you can add and multiply them, and as long as they aren't zero, then they have an inverse. thanks!2017-11-29