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Questions tagged [separable-extension]

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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 ...
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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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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 \...
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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 $F$ ...
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1answer
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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 ...
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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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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 ...
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1answer
147 views

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 $x^p-\alpha^p=(...
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$\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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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 $\iff\gcd(f, Df) = 1$....
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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 ...
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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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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
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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
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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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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?
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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 ...
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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 ...
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0answers
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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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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 ...
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1answer
151 views

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 $\mathbb{F}_q$. ...
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 ...
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 $f=[(x-u_1)\cdots (x-u_n)]^{[K(u_1):K]_s}$ ...
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1answer
52 views

$x^2-3$ is separable over $\mathbb Q$ but not separable over $F_2$

$x^2-3=(x-\sqrt{3})(x+\sqrt{3})$ over $\mathbb Q$, so that part makes sense. Now, when it says $x^2-3$ is a polynomial over $F_2$, I imagine it means all the coefficients are calculated mod $3$, so $...
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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 ...
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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 ...
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0answers
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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 ...
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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 $F$. ...
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: ...
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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?
2
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1answer
599 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. ...
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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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3answers
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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 $\...
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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 [\...
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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 ...
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2answers
86 views

Proving that the polynomial $f(x)$ is separable

Let $f(x)=x^{4}-10x^{2}-25\in \mathbb Q[x]$. I want to prove that $f(x)$ is a separable polynomial over $\mathbb Q$. I know that from the definition $f(x)$ is separable if none of the irreducible ...
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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 ...
1
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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 ...
1
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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\...
1
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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 $\{...
1
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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 ...
2
votes
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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2answers
84 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 ...
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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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) \...