# Questions tagged [separable-extension]

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74 questions
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### If Gal(f/K) is simple then for each extension F/K, Gal(f/F) = Gal(f/K) or trivial.

I have a version of natural irrationalities theorem that states: Let $F/K$ be a extension of fields. Let's denote by $Gal(f/K)$ the Galois group of the extension $K(\alpha_1,\ldots,\alpha_n)$ for ...
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### Simple extension is splitting field of $f=X^p-X+a \in K[X]$ with $char(K)=p$

Given is the polynomial $f=X^p-X+a \in K[X]$, where $K$ is a field of characteristic $p$ and $a \in K$. My task now is to show that every simple extension of $f$ is already the splitting field of $f$....
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### Does $L_1(w)=L_2(w)$ imply $L_1=L_2$?

Let $L_1 \subseteq L_2 \subsetneq L_3$ be inclusions of three fields, where $L_1$ may or may not equal $L_2$, and $L_2$ is strictly contained $L_3$. Assume that the three field extensions are finite ...
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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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### “Intersection” of separable subfields [duplicate]

I have the following question, from Isaacs' Algebra book. Suppose $F\le E$ is a finite-degree normal field extension, and that $K$ and $L$ are intermediate subfields (between $F$ and $E$). If $E$ is ...
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### How to prove this ‘lemme connu’?

At the end of Exposé I in SGA 1 it is asserted that a well known lemma states the following: Let $k$ be an infinite field and $E$ be a finite product of finite field extensions of $k$. Suppose not ...
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### Interesting algebras over non-commutative rings

It would be nice to have several examples of an interesting $R$-algebra $A$, where $R$ is a non-commutative ring (plausible definitions can be found here). One example is a polynomial ring over any ...
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### Proving a sufficient condition for the separability of a finite extension over a field of non-zero characteristic

I'm taking an intro to Galois theory course, which is rather exciting. We had the following question in a practice paper: Let $K$ be a field of characteristic $p$, and let $L/K$ be a finite ...
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### Let $D'$ the subring of $D$-integral elements of $E$. Why $D'$ is Dedekind?

I am studying the theorem: Let $D$ be a Dedekind domain, $F$ its field of fractions, $E$ a finite dimensional extension field of $E$, $D'$ the subring of $D$-integral elements of $E$. Then $D'$ is ...
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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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### 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 ...