Questions tagged [separable-extension]

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74 questions
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Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.

Let $K$ be the rational function field $k(x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u\in K$, and write $u = f(x)/g(x)$ with $f$ and $g$ relatively prime. ...
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Automorphism group of separable closure

Let $k$ be a field and $k_s$ its separable closure. I would like to understand why $\mathrm{Aut}_k(k_s)$ is an inverse limit of the groups $\mathrm{Gal}(L/k)$, where $L$ is a finite Galois extension ...
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(From Milne) If $L/F$ is an extension of ﬁelds of degree 2, then there is an automorphism $\sigma$ of $L$ such that $F$ is the ﬁxed ﬁeld of $\sigma$

This is an exercise in Milne's notes. The answer is short, it says: (a) is false—could be inseparable. (b) is true—couldn’t be inseparable. So may I please ask how does it related to separablity? ...
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Questions regarding tower of normal/separable extensions

I am learning about Galois theory these days. And I am considering to prove: Is that the fact that given a tower of extensions $A/B/C/D$, if $A/B$, $B/C$, $C/D$ are normal, then $A/D$ is normal? Is ...
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Example for non separable elements?

We say a polynomial $P \in \mathbb{K}[X]$ is seperable (where $\mathbb{K}$ is a field) if and only if $P$ has only simple roots in the algebraic closure of $K$. We say an element $x$ is seperable if ...
Let $F\subseteq L\subseteq K$ be fields such that $K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal.
While studying Patrick Morandi's book "Field and Galois Theory", on page49, I came across the following question: Let $F\subseteq L\subseteq K$ be fields such that $K/L$ is normal and $L/F$ is purely ...