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I have the doubt that is any finite extension included in Galois extension? link. This link states that case on finite field and I understand it. But what about the field of characteristic $0$ and infinite field of characteristic $p$ ,I haven't found any content online and now I am very confused. Could you give me a picture of relationships between these two kinds of extension in different situations

Tommy Xu
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    A finite extension $M/K$ is contained in a Galois extension $L/K$ if and only if it is separable. Separability is clearly a necessary condition. Try to prove it is also sufficient. – Mark Sep 25 '24 at 15:31
  • @Mark, Can I do like this. For extension $L/M/K$, $L/K$ is Galois and $M/K$ is finite, then L is a splitting field of a separable polynomial $f$ over $K[x]$. And $M$ is $K(a_1,a_2, . . . a_n)$, so for each a_i, it belongs to $L$ then is a root of $f$. In sum, this finite extension is separable. – Tommy Xu Sep 26 '24 at 04:19

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