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Questions tagged [separable-extension]

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5
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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 $$ K_s = \left\{ {x \in L \mid x \text{ is algebraic and separable over } K} \right\} $$ ...
3
votes
1answer
71 views

Converse to Prop. V.6.11 from Lang's *Algebra*: $E/k$ normal $\impliedby E^{\operatorname{Aut}(E/k)}/k $ purely inseparable?

In Lang's Algebra, he proves in Proposition 6.11 of Chapter V (page 251, third edition) the decomposition of a normal extension into a tower of a purely inseparable extension followed by a separable ...
1
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1answer
21 views

Is the degree of an irreducible inseparable polynomial always a p-power?

Consider a field $K$ of characteristic $p>0$. Let $f$ be an irreducible, inseparable polynomial in $K[Y]$. I'm wondering if the degree of $f$ has to be a power of $p$. For instance, the standard ...
0
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1answer
35 views

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 $f=[(x-u_1)\cdots (x-u_n)]^{[K(u_1):K]_s}$ ...
0
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1answer
49 views

Show that $\mathbb{Q}[\sqrt{2},\sqrt{3}] $ is separable

how can I characterize the minimal polynomials of all elements of this extension 'to show that they have all the distinct roots?
0
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1answer
72 views

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 ...
0
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1answer
92 views

$k$ 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),...
0
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1answer
43 views

$L/k$ 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 ...
7
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0answers
131 views

What is an example of a non-simple finite extension $K/F$ such that the purely inseparable closure of $F$ in $K$ is simple?

The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely ...
2
votes
0answers
18 views

A Potential Relation Implied by Separability

Upon reviewing concepts related to separable extensions, I came acress the following (seemingly) nontrivial implication. Perhaps I have made a trivial mistake somewhere. Here it is: Let $K$ be a ...
2
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0answers
48 views

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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0answers
32 views

What sufficient criteria are there for a polynomial to be separable?

I know that $p$ is separable when the discriminant is not zero, $p'$ and $p$ don't have a common root, $p$ is irreducible, but that's it. Is there a criterion over $\mathbb Q$ for separability in ...
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0answers
31 views

Surjectivity of multiplication by $n$ on elliptic curves

Is there an abelian variety $A$ over a field $k$, such that $A(k^{\rm sep})$ is not a divisible group? The motivation of my question is the following : if $L$ is any algebraically closed field, then $...
1
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0answers
7 views

$L_1L_2/K$ 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 ...
1
vote
0answers
65 views

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable. I am really unsure how to solve this. I know that $K/F (\alpha) $ is ...
1
vote
0answers
44 views

$\mathbb F_p(t) \leq L $ a finite extension

Given the extension in the title is finite, show that there exists an $n\geq 0 $ and $y \in L$ such that $y^{p^n} = t$ and $\mathbb F_p(t)(y) \leq L$ is a separable extension. I have currently the ...
1
vote
0answers
94 views

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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0answers
6 views

$\mathbb{F}_{p}(t)$ separable over $\mathbb{F}_{p}(f(t)/g(t))$ when $\deg(f), \deg(g) < p$.

Let $f(t), g(t) \in \mathbb{F}_{p}[t]\setminus \{0\}$ where $t$ is an indeterminate, and where $\max\{\deg(f), \deg(g)\} < p$ and $f(t)/g(t) \not\in \mathbb{F}_{p}$. Show that the extension $\...
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0answers
5 views

Maximal separable subalgebras of semisimple associative algebras

Is anything known about maximal separable subalgebras of semisimple associative (and not necessarily commutative) algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic ...
0
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0answers
85 views

Equivalent definitions of separable extension of a field

Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that: $F$ is said to be separable over $\boldsymbol{k}$ if it satisfies the following equivalent conditions ($p$ denotes the ...
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0answers
33 views

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?
0
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0answers
25 views

$Gal(f)$ 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 ...
0
votes
0answers
10 views

Separable polynomial with repeated roots.

Let $E/F$ be an extension and $f \in F[X]$. I was reading a statement saying if $f$ is separable and has multiple roots (repeated roots), these roots must be in $F$. I don't understand this. Isn't the ...
0
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0answers
42 views

If $a \in L-k $ satisfies $k(a^n)=L$ (for all $n \geq 1$), then $L/k$ is Galois?

Let $k \subsetneq L$ be a finite separable field extension, and let $a \in L-k$ satisfy: For every $n \geq 1$, $k(a^n)=L$. In other words, all the non-zero powers of the primitive element $a$ are ...
0
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0answers
26 views

Problem regarding proving an extension of a field to be separable

The whole question looks like- Let, $x^p-x-1$ be a polynomial over a field $F$ of characteristic $p\ne 0$ and $\alpha$ be a root of it. Prove that $F(\alpha)$ is separable extension over $F$. ...
0
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0answers
114 views

If $K $ is infinite and $u,v $ are separable and algebraic over $K $ then there is some $a \in K $ such that $K (u+av)=K (u,v) $ proof verification

Let $K $ be a field with infinite elements. Prove that if $u,v $ are separable and algebraic over $K $ then there is some $a \in K $ such that $K (u+av)=K (u,v) $. Is the result still true if $K$ is ...