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Questions tagged [separable-extension]

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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 ...
5
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2answers
511 views

$F/K$ 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 ...
5
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1answer
83 views

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 ...
5
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2answers
61 views

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 $F$ ...
5
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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 $$ K_s = \left\{ {x \in L \mid x \text{ is algebraic and separable over } K} \right\} $$ ...
4
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1answer
2k views

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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3answers
94 views

A perfect field that is neither of characteristic $0$ nor algebraically closed

Just recently I was given the task to find a non separable field extension. At first it seems like an easy task, but common fields you encounter are usually perfect. Eventually I found the example $\...
3
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2answers
50 views

Is it true that $Gal(K/F)\cong S_{n_1}\times \cdots S_{n_k}$?

I was reading galois theory and galois group from Dummit Foote and while reading Galois groups of polynomial a sudden question came into my mind that if $f(x)$ is an irreducible separable polynomial ...
3
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2answers
53 views

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 ...
3
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1answer
77 views

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 ...
3
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1answer
61 views

Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite?

Let $E/K$ be an algebraic extension of fields and denote by $E_s$ the separable closure of $K$ in $E$, i.e. the set of all elements of $E$ that are separable over $K$ (which can be shown is a (...
3
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1answer
145 views

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, ...
3
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1answer
173 views

“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 ...
3
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1answer
98 views

Sufficient conditions for vanishing module of Kähler differentials

It is a standard fact that the module of Kähler differentials $\Omega_L$ of a finite separable extension of a field $k$ is equal to 0. I also know that for a ring B and a finite extension $k \...
3
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1answer
146 views

If $L/K$ normal and $H = \operatorname{Aut}(L/K)$, then $L/L^H$ is separable and $L^H/K$ is purely inseparable.

I need to prove the following: Let $L/K$ be a normal field extension. Denote by $H=\operatorname{Aut}(L/K)$ the Galois group of the extension, and by $L^H$ the fixed field of $H$ in $L$. Prove that ...
3
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1answer
123 views

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 ...
3
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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 ...
2
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2answers
185 views

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 ...
2
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1answer
601 views

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. ...
2
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1answer
478 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 ...
2
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2answers
82 views

Non-Separable Polynomials and their Derivatives

We say that a polynomial $f(x) ∈ F[x]$ is inseparable if it has a repeated root in some field extension. Otherwise we say that $f(x)$ is separable. Prove that $f(x)$ is separable $\iff\gcd(f, Df) = 1$....
2
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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 ...
2
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2answers
80 views

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 ...
2
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1answer
65 views

Let $L:K$ be a Galois extention, show that $L:M$ is a normal.

Assume the field extension $L:K$ is Galois (i.e. finite, normal and separable), with $M$ an intermediate field. Show that $L:M$ and $M:K$ are finite separable field extensions. Attempt: ...
2
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2answers
80 views

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
135 views

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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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
65 views

Prove this field extension is separable [duplicate]

Let $F \subset G_i \subset E, (i=1,2)$ where $E$ is normal and finite dimensional over $F$. Assume $E$ is separable over $G_1$ and $G_2$. Prove $E$ is separable over $G_1 \cap G_2$ There is a hint ...
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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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2answers
85 views

Let $L/K$ be a finite separable field extension. How to prove there are finitely many intermediate fields?

By the Primitive Element Theorem, $L=K(\alpha)$ for some $\alpha \in L$. So, there is a $K$-basis for $L$ given by powers of $\alpha: 1, \alpha,...,\alpha^{n-1}$ where $n$ is the degree of the ...
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1answer
125 views

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
26 views

An Upper and Lower Bound for a Field Extension of $\mathbb{Q}$ by a Complex Root

Consider the polynomial $f(x) = x^3 + \zeta x + \sqrt{3}$ in $\mathbb{C}[x],$ where $\zeta$ is a primitive third root of unity. Given a root $\alpha$ of $f(x)$ in $\mathbb{C},$ prove that $4 \leq [\...
1
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1answer
41 views

If $E/k$ is not separable, but satisfies the property that there exists a positive integer $n$ for which every $\alpha \in E$ of degree less than $n$

In chapter V of Lang's graduate algebra, there is a Lemma 1.7 saying that if $E/k$ is separable and satisfies the property that there exists a positive integer $n$ for which $\alpha \in E$ has degree $...
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1answer
112 views

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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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
51 views

if the tensor product of a finite field extension and the separable closure over base field is a field, then their intersection is the base field

This is a step in Corollary 3.3.21 in Liu (Algebraic geometry and Arithmetic curve) In the proof, we have $K/k$ a finite extension, and $k^s$ the separable closure of $k$. And we showed that $K\...
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1answer
125 views

Suppose $K/F $ is finite with $K = F (K^p) $. If a finite set $S $ is linearly independent, prove that so is $S^p $.

Suppose that $F $ is a field with characteristic $p >0$, that $K/F$ is a finite extension, and that $K=F (K^p) $. If $\{x_1,...,x_n \} \subset K $ is linearly independent over$F$, then so is $\{...
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1answer
281 views

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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1answer
58 views

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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1answer
25 views

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
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 ...
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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 $...
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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 ...
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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 ...
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1answer
38 views

Question about separable polynomial - proof verification

Let $K/F $ be a separable extension with char $F =p > 0$. Prove that for any given $\alpha \in K , \alpha \in F (K^p) $. I am not sure how to approach this problem. I think Viete's formula is too ...
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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 ...
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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 $f$....
0
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2answers
46 views

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) \...
0
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1answer
48 views

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 ...