# Questions tagged [separable-extension]

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### Extension over the intersection of intermediate fields is also separable [duplicate]

Suppose $L|F$ is a finite Algebraic, normal extension. Let, $F \subset K_1, K_2 \subset L$ be two fields such that $L|K_1 , L|K_2$ are Algebraic and separable. Show that $L | K_1 \cap K_2$ is also a ...
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### Field is perfect iff every element has pth root.

I am trying to understand the proof for one direction of the following theorem: I am confused about the part in red. Why does this work? Has it something to do with the Frobenius endomorphism?
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### Help understanding how to show a field extension is a Galois extension .

I have the field extension $\Bbb Q(\sqrt[8]{2},i)$ over $\Bbb Q$. I want to show that this is a Galois extension. I know that I can do this by showing that It is an extension which is both normal and ...
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### $Gal(f)$ is cyclic of order $m$ for irreducible $f \in K[X]$ of degree $m$, where $K$ is a finite field

Let $K$ be a finite field, $f \in K[X]$ irreducible with degree $m$. Show that $Gal(f)$ is cyclic of order $m$. I have shown that $f$ is separable over $K$ by using that $K$ is finite and thus ...
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### $L_1L_2/K$ is separable. justify it

Is it true that- If $L_1/K$ and $L_2/K$ are extensions contained in a field $F$ and both are separable then $L_1L_2/K$ is separable. If not true then give me any counter example. Answer: In ...
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### Is this a counterexample?

Suppose $K$ is a field and $\overline K$ an algebraic closure. Let $f$ be a $K$-automorphism of $\overline K$, let $L$ be the subfield of $\overline K$ fixed by $f$. In this post : (link), they ...
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### Lang Steinberg over separably closed field

Let $K=K^{sep}$ be a separably closed field with $K|\mathbb{F}_q$, where $\mathbb{F}_q$ is the field with $q$ elements. Let $\mathbb{G}$ be a connected linear algebraic group over $\mathbb{F}_q$. ...
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### Proof of separability of polynomials without derivatives

Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.) EDIT: As people ...
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### The multiplicity of a root $r$ of a irreducible polynomial is a power of $p$ characteristic

$f$ is an irreducible polynomial over a field $K$ of characteristic $p$. $F$ is a splitting field of $f$ over $K$ and $u_1$ a root of $f$. I have shown that $f=[(x-u_1)\cdots (x-u_n)]^{[K(u_1):K]_s}$ ...
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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 $f$....
### 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 ...
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 ...