Is there a field extension $L/K$ such that it is an infinite algebraic extension of fields but the separable degree of $L$ over $K$ is finite?
2026-04-22 16:08:11.1776874091
About extension of fields
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Yes there are infinite purely inseparable algebraic extensions. Take for example $K=\mathbb{F}_p(X)$ and $L=\mathbb{F}_p(X^{1/p^{\infty}})= \varinjlim_n ~ \mathbb{F}_p(X^{1/p^n})$.