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All finite fields (they have char $p$ for some $p$ prime) are perfect. A well known field which is not finite and not perfect with char $p$ is $\mathbb{F}_p(t)$, the field of rational functions with values in $\mathbb{F}_p$ with an unknown $t$.

Can someone show me an example of perfect, infinite field with char $p$?

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A trivial example is the algebraic closure of $\Bbb F_p(t)$.

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    $\begingroup$ Or more generally every algebraically closed field. $\endgroup$ – Paul K Dec 30 '19 at 14:29
  • $\begingroup$ Well.. yes of course it follows by definition. But "what kind of elements are there"? And how to find any other non-trivial example? $\endgroup$ – Gabrielek Dec 30 '19 at 14:29
  • $\begingroup$ @Gabrielek how about algebraic closure of $\mathbb{F}_p$? It is perfect and you can describe all its elements. $\endgroup$ – Slup Dec 30 '19 at 14:30
  • $\begingroup$ @Paul K $\mathbb{C}$ is algebraically closed over $\mathbb{R}$ but it has char $0$ $\endgroup$ – Gabrielek Dec 30 '19 at 14:30
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    $\begingroup$ $\bigcup_n \Bbb{F}_p(t^{1/p^n})$ is easier to visualize $\endgroup$ – reuns Dec 30 '19 at 14:33

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