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Questions tagged [separable-extension]

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Is $\mathbb{F}_q(x)$ a perfect field?

It is well known that the finite fields $\mathbb{F}_q$ are perfect i.e. every finite extension of $\mathbb{F}_q$ is separable. I am wondering if a pure transcendental extension $\mathbb{F}_q(x) \...
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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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non separable, non normal, finite field extension

I want to give an example of a non-separable, non-normal, finite field extension. So I have to find an extension of a field, which is not perfect. I would suggest $K=\mathbb{F}_2(t)$. An extension, ...
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If $\alpha$ separable over $F$ then $F(\alpha )/F$ is a separable extension.

Let $K/F$ be a field extension and $\alpha \in K$ is algebraic over the field $F.$ Now suppose $\alpha$ is separable over $F.$ Then how can I show that $F(\alpha)/F$ is a separable extension, i.e., an ...
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Find the degree of splitting field of a separable polynomial over finite field

Question: Let $f$ be a polynomial of degree $n$ over finite field $F=\mathbb{F}_q$ and $E$ be the splitting field of $f$. If $f$ has $n$ simple roots over $E$, can we say that $[E:F] \mid n$ ? ...
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If $Gal(f/\mathbb{Q}) \cong S_n$ then $Gal(f/\mathbb{Q}(\alpha_1)) \cong S_{n-1}$?

Suppose I denote by $Gal(f/\mathbb{Q})$ the Galois group of the extension given by the splitting field of a separable polynomial $f$ of degree 6 over field $\mathbb{Q}$. Suppose that, $Gal(f/\...
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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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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/...
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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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(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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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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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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$L/k$ 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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$k$ 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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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 ...