Given a perfect field of prime characteristic $p$, is it necessarily finite? I believe there must be some counterexample. However, the only infinite field of characteristic $p$ that I know of is $\mathbb{Z}_p(t),$ which is not perfect. Any hints?
2026-03-28 12:32:16.1774701136
Infinite perfect field of characteristic p
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You can take, for instance, the field $\mathbb F_p(x,x^{1/p},x^{1/p^2},\ldots)$.