Galois theory-some concepts

xyz5261941

The Galois theory on finite algebraic extensions of perfect fields are fairly well understood(at least for me). Yet, when it comes to imperfect fields, things begin to get complicated. I write this post mainly to clarify these closely related, yet not so well distinguished concepts(at least for myself).

First of all, what is a perfect field? Before talking about that, perhaps it is proper to ask what it means by a polynomial being separable. Suppose that $latex k$ is a field, and $latex P(X)in k[X]$ is an irreducible polynomial. So, we say that $latex P(X)$ is separable if it has no multiple roots in an(and hence in any) algebraic closure $latex bar{k}$ of $latex k$. There is an equivalent way to decide if $latex P(X)$ is separable or not, that is, take the formal derivative $latex P'(X)$ of $latex P(X)$ and see if $latex P(X)$ and $latex P'(X)$ are coprimeā€¦

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