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Questions tagged [separable-extension]

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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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If $K$ is a field extension of $F$ and if $\alpha\in K$ is not separable over $F$, show that $\alpha^{p^m}$ is separable over $F$ for some

If $K$ is a field extension of $F$ and if $\alpha\in K$ is not separable over $F$, show that $\alpha^{p^m}$ is separable over $F$ for some $m\geq 0$, where $p = $char$(F)$. I know that $x^p-\alpha^p=(...
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1answer
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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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1answer
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Separable algebras over a non-commutative ring

What is the 'correct' definition of a separable algebra over a non-commutative ring? Are there known results about such algebras? Examples? Recall that one of the equivalent definitions of a ...
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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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1answer
32 views

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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1answer
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Can I found $x$ such that $K$ is a separable over $\textbf{F}_q(x)$?

Let $q$ be a power of a prime number, $K$ be a field containing $\textbf{F}_q$ and of transcendance degree one over it. Can I found $x\in K$ such that $K$ is a finite and separable field extension of $...
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1answer
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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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1answer
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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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1answer
49 views

How can I see that $\Bbb F_{p^2}(t)$ is a separable extension of $\Bbb F_p(t)$?

I am given 2 examples to see that a imperfect field can have both separable and inseparable extensions. I am told that $\Bbb F_{p^2}(t)$ is a separable extension of $\Bbb F_p(t)$, where $\Bbb F_p(...
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2answers
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Obtaining invertible matrix from a separable field extension

I was reading a proof in Bourbaki, Chapter VIII on Non-commutative algebra ($\S7$, n$^{\circ}2$, Proposition $3$ b)). In the proof, they claim the following result (translated from French) : Let $A/...
2
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1answer
481 views

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 ...
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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 ...
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1answer
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$k$ be perfect field , char $p>0$ , $u=f(X)/g(X) \in k(X) ; f(X),g(X) \in k[X]$ relatively prime , $k(X)/k(u)$ separable ; to show $u \notin k(X)^p$

Let $k$ be a perfect field of characteristic $p>0$ ( perfect means $k^p:=\{a^p : a \in k\}=k$ ) . Let $E=k(X)$ be the rational function field over $k$ , let $u=f(X)/g(X) \in E=k(X)$ such that $f(X),...
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$L/k$ finite extension , $L_1,L_2 $ subfields of $L$ containing $k$ , $L_1/k$ separable and $L_2/k$ normal , then $[L_1L_2:L_2]=[L_1:L_1\cap L_2]$ ?

Let $L/k$ be a finite extension . $L_1,L_2 $ subfields of $L$ containing $k$ such that $L_1/k$ is separable and $L_2/k$ is normal . Then it is easy to see $L_1L_2/L_2$ is separable . But how to show ...
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1answer
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If $[F:K] = n$ and there exist $n$ monomorphisms $\sigma_i \colon F \to \bar{K}$, then is $F/K$ separable? [closed]

Let $F$ and $K$ be two fields, such that $F$ is a finite extension of $K$, and $[F:K]=n$. Assume that there exist exactly $n$ monomorphisms $\sigma_1,\sigma_2,\ldots,\sigma_n$ $$\sigma_i: F\...
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If $F^p = F$ and $E/F$ is algebraic, then $E/F$ is separable and $E^p = E$ : Corollary V.6.12 from Lang's *Algebra* [duplicate]

This is from Lang's Algebra (page 251) Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $\operatorname{Aut}(E/F)$. Then, $E^G$ is purely inseparable over $F$ ...
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Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
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1answer
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The defintion of the the separable closure of a field

Could someone tell me the definition of the separable closure of a field $K$? Furthermore, I would like to know whether it is a Galois extension of $K$? Also, why is this construction useful?
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2answers
512 views

Show that if $K \subset L$, then the separable closure of $K$ in $L$ is a field

Let $K \subset L$ be a field extension. Consider the separable closure $K_s$ of $K$ in $L$ defined as $$ K_s = \left\{ {x \in L \mid x \text{ is algebraic and separable over } K} \right\} $$ ...
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$F/K$ finite extension, then $\exists !$ intermediate field $K \subset L \subset F$ such that $L/K$ separable and $F/L$ purely inseparable

I've completed an introductory course in Galois Theory, but feel my understanding of separability is poor. I think my confusions boil down to the following question: What is the relationship ...