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Mathematics

Questions tagged [separable-extension]

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64 questions
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1answer
89 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 \...
0
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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 $\...
2
votes
2answers
76 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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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 ...
1
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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 ...
0
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0answers
84 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
30 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 $...
0
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0answers
18 views

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 ...
0
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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
votes
1answer
47 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
24 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 ...
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 ...
0
votes
1answer
150 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
votes
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
votes
1answer
32 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
votes
1answer
50 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 $...
3
votes
2answers
49 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 ...
0
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0answers
9 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 ...
3
votes
1answer
72 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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0answers
6 views

Question regarding separable extension

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 have been trying it for ...
0
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0answers
41 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
24 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
votes
1answer
48 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?
3
votes
3answers
87 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 $\...
1
vote
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 [\...
2
votes
1answer
63 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: ...
0
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0answers
105 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 ...
0
votes
2answers
69 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 ...
1
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0answers
63 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
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
48 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
vote
1answer
120 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
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 ...
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 ...
1
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2answers
73 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
38 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 $...
2
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0answers
22 views

$\left[E:K\right]_s < \infty \implies \left[E_s:K\right] < \infty$

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 (...
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) \...
3
votes
1answer
134 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, ...
1
vote
1answer
267 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 ...
0
votes
1answer
108 views

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 $[E:F] \mid n$ ? ...
1
vote
1answer
111 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/\...
0
votes
1answer
39 views

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 ...
1
vote
0answers
92 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 $f$....
3
votes
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 ...
1
vote
1answer
543 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. ...
0
votes
1answer
69 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
votes
1answer
145 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=(...
3
votes
1answer
171 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 ...
2
votes
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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