Let $E$ be a local field of characteristic $p>0$. Let $\varphi$ be the absolute Frobenius map on $E$. Then how to prove that the field extension $E/\varphi(E)$ is purely inseparable.
2026-03-25 22:09:40.1774476580
Why this extension is purely inseparable.
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It doesn't matter that it is a local field. Each $a \in E$ is the root of the purely inseparable polynomial $$(x-a)^p = x^p - a^p \in E^p [x]$$