Questions tagged [separable-extension]
The separable-extension tag has no usage guidance.
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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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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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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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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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1answer
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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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1answer
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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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1answer
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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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$x^2-3$ 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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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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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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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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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 ...
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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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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 $F$.
...
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1answer
45 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?
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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
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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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1answer
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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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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
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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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49 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
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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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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
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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 $\{...
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$\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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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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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
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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 $...
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$\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 (...
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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) \...
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1answer
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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, ...
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1answer
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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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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 $[E:F] \mid n$ ? ...
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1answer
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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
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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 $f$....
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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 ...
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1answer
446 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
65 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 ...
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1answer
120 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
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1answer
163 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 ...
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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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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
118 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 ...
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2answers
70 views
Proving a sufficient condition for the separability of a finite extension over a field of non-zero characteristic
I'm taking an intro to Galois theory course, which is rather exciting. We had the following question in a practice paper:
Let $K$ be a field of characteristic $p$, and let $L/K$ be a finite
...
