0
$\begingroup$

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 $p$?

Improve this question
$\endgroup$
4
$\begingroup$

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

Improve this answer
$\endgroup$
  • 2
    $\begingroup$ 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
  • 2
    $\begingroup$ $\bigcup_n \Bbb{F}_p(t^{1/p^n})$ is easier to visualize $\endgroup$ – reuns Dec 30 '19 at 14:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.