Questions tagged [separable-extension]
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155
questions
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1answer
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Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$.
Question: Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$ for some positive integer $m$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$. (I think we also ...
2
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1answer
48 views
Is an algebraic extension of a separably closed field separably closed?
In the book "Rational points on varieties" by Bjorn Poonen, exercise 1.1 (a) states:
"Prove that an algebraic extension of a separably closed field is
separably closed."
Is this ...
1
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0answers
62 views
Equality between degree of separability of field extensions.
The Problem: Let $k \subset F \subset L$ such that $[L:k] < \infty.$ Let $S_1$ be the separable closure of $k$ in $F$, $S_2$ the separable closure of $F$ in $L$ and $S$ be the separable closure of $...
1
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0answers
43 views
Functional Analysis : A question abaout extensions without use the Banach Theorem or Zorn's lemma . [duplicate]
The question is :
Let be $X$ a separable normed vectorial space and $S$ a subspace de $X$ $ \varphi : S \rightarrow \mathbb{F}$. Show without use Hahn- banach' Theorem (and zorn's lemma) that exists ...
0
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1answer
33 views
Is every isogeny over $\mathbb Q$ separable?
I am reading a proof of a simplified version of the weak Mordell-Weil theorem, where we only consider elliptic curves over $\mathbb Q$. Now, in the proof, they mention some (non-constant) isogeny, and ...
0
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1answer
38 views
Is the proposition $[KL:L]=[K:F]$ iff $K\cap L=F$ for finite Galois extension still hold for separable extension?
I know that when $K/F$ and $L/F$ are finite Galois extension (where $L, K<\overline{F}$), $[KL:L]$ divides $[K:F]$, and $[KL:L]=[K:F]$ iff $K\cap L=F$.
My question is that above proposition is ...
1
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0answers
54 views
Every algebraic extension of a finite field $F$ is separable [duplicate]
Question: I want to show that every algebraic extension of a finite field $F$ is separable.
Thoughts: I think I know how to do this: let $F\subseteq E$ be an algebraic extension of $F$. Let $a\in E$. ...
0
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1answer
31 views
How to reduce to the affine case? Number of points in the fibre and the degree of field extension.
Let $f : X\to Y$ be a dominant morphism of integral algebraic varieties over $\mathbb C$. Suppose $[K(X): K(Y)]=n$. Then there exists a dense open subset $U$ of $Y$ such that $f^{-1}(y)$ consists in $...
0
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0answers
22 views
Inseparable field extension of degree 2
I have searched for an example of a degree 2 field extension that is not separable.
The example I see is the extension $L/K$ where $L=\mathbb{F}_2(\sqrt t), \ K=\mathbb{F}_2(t) $ where $t$ is not a ...
0
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0answers
23 views
Unseparable element becomes separable
Let $L/K$ be an extension of fields. Denote by $p$ the characteristic of $K$. One assumes that $p$ is a a prime. Let $a\in L$ not separable over $K$. Does it exist an integer $s$ such $a^{p^s}$ is ...
0
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0answers
18 views
Normal closure as the compositum of conjugates
An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242.
Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
1
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1answer
44 views
Theorem 17.22 from Ian Stewart's Galois Theory
Apply the Frobenius map to minimal polynomials to see that $$ [K(\alpha^p+\beta^p):K(\alpha^p+\beta^p,\beta^p)]\leq [K(\alpha+\beta):K(\alpha+\beta,\beta)] $$ and $$ [K(\alpha^p+\beta^p):K] \leq [K(\...
2
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0answers
33 views
For which $p$ is $\mathbb F_p(X)$ separable over $\mathbb F_p(X^6)$?
Here's what I've got, but I'm not confident it's airtight...
Let $K=\mathbb F_p(X^6)$ then $\mathbb F_p(X)=K(X)$ and the polynomial $F(T) = T^6-X^6\in K[T]$ has $X$ as a root. It's irreducible by ...
3
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0answers
73 views
$f(x)=x^4+x+1$ is separable over $\Bbb F_p$ iff $p=229$
I was asked to prove this in my homework, but when I tried looking at the roots of this polynomial over $\Bbb F_{229}$ I noticed $75$ is a double root of this polynomial and $f=(x-75)^2(x^2+150x+158)$....
1
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1answer
86 views
$E$ is the splitting field over $F$ for some separable $f(X) \in F[X]$ implies that $f(X)$ is separable over $\Phi(\Gamma(E/F))$?
I'm going over the proof of Theorem 25.1 in Abstract Algebra, and I'm trying to understand a particular detail concerning separable polynomials. Here's the relevant portion of the theorem statement:
...
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0answers
25 views
Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal
Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
1
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0answers
45 views
Properties of $k[x(x-1)]_{\langle x(x-1) \rangle} \subseteq k[x]_{\langle x \rangle}$
Let $k$ be aa arbitrary field.
Let $R=k[x(x-1)]_{\langle x(x-1) \rangle}$ and let $S=k[x]_{\langle x \rangle}$,
$m=x(x-1)R$, $n=xS$, $k(m)=R/m$, $k(n)=S/n$.
We have, $mS = n$ (since $x-1$ is ...
0
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0answers
18 views
Separable algebraic extension and Jacobian determinant
Let $\{x_1, \cdots, x_n\}$ be the set of elements in some extension of field $F$. If there are $n$ multivariate polynomials $f_i(z_1, \cdots, z_n)\in F[z_1, \cdots, z_n]$ ($i=1,\cdots,n$) such that $...
3
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1answer
76 views
Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?
I am studying these notes and I am trying to generalize a bit the setting of the Section 3, because there doesn’t seem to be a fundamental reason to only study $p$-adic fields. So all the fields ...
0
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1answer
22 views
Separability degree of extension as number of intermediate separable extensions.
I read in another SE post that we can think of separability degree in this way but I can't explain why. Specifically, given an algebraic extension L/K, I want to understand this to see why the ...
2
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1answer
33 views
Ring $R'$ with $R\subset R'\subset T$ where $R$, $T$ integrally closed and obtained by a purely inseparable extension
Let $R$ be an integrally closed domain with field of fractions $K$. Let $M/K$ be a purely inseparable extension and let $T$ be $R$'s integral closure in $M$. Consider a subring $R\subset R'\subset T$ ...
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0answers
54 views
Why does separable degree behave unexpectedly in infinite extensions?
First of all, my definition of separable degree of $K/F$ is $[K:F]_s=|\operatorname{Hom}_{F-alg}(K,\bar F)|$ where bar denotes algebraic closure.
It is well-known that in finite field extensions $[K:F]...
-3
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2answers
90 views
prove f is separable if and only if f is relatively prime to its derivative. [closed]
Given the definition; a polynomial f in the complex numbers is separable if it has only simple roots.
My question is how do I prove f is separable if and only if f is relatively prime to its ...
2
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0answers
57 views
What is the intuition behind the definiton of a seperable field extension?
so I am currently studying separable extensions in John Fraleigh's abstract algebra book and have come upon this definiton of a separable field extension:
A finite extension $E$ of $F$ is a separable ...
0
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0answers
14 views
proving a finite field extension separable
$F/K$ be a finite field extension and $F=K(F^p)$. Then show that $F/K$ is separable extension.
I have been asked to solve the above problem. The problem seems to miss some information. It must be ...
3
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1answer
34 views
Converse of the result about the composite of purely inseparable extensions
This post (Composite of two purely inseparable extensions is purely inseparable.) says that if $E < F$ is purely inseparable and $F < K$ is purely inseparable, then $E < K$ is separable.
...
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0answers
64 views
Separability of $\mathbb{C}+ \langle h \rangle \subseteq \mathbb{C}[x]$
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
Now ...
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0answers
14 views
How to find degree of separability and degree of inseparability in following question
This question is from an abstract algebra assignment which I am trying.
Let char K = p $\neq 0$ and let $f\in K[x]$ be irreducible of degree n. Let m be the
largest nonnegative integer such that f is ...
0
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1answer
40 views
If $\operatorname{char} K =p$ and $[F:K]$ is finite and not divisible by $p$, then $F$ is separable over $K$
Consider the following question,
If $\operatorname{char} K =p$ and $[F:K]$ is finite and not divisible by $p$, then prove that $F$ is separable
over $K$.
I am really sorry but I am not able to use ...
0
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1answer
28 views
A question based on intermediate fields , separable elements , inseparable elements
Please consider the following question:
Let F be an algebraic extension field of K , S the set of all elements of F
which are separable over K, and P the set of all elements of F which are purely ...
0
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1answer
38 views
Proving that polynomial f must be of this form if the polynomial is given to be separable
If $f \in K[x] $ is monic irreducible, $\deg (f) \geq 2$, and has all its roots equal (in a splitting
field), then $\text{char }K = p \neq 0$, and $f = x^{p^n} - a$ for some $n\geq 1$ and $a\in K$...
1
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1answer
20 views
One element is given seperable and another purely inseperable then show that …
Kindly help with the following question asked in my mid term exam which has been concluded.
If u $\in F$ is separable over K and c $\in F$ is purely inseparable over K, then
K(u,c) = K( u + c).
I am ...
0
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0answers
28 views
Given F is Galois over E and E is Galois over K and a splitting field condition is given
Consider the following problem: This question is from Thomas Hungerford Section 3 Problem 15.
If F is algebraic Galois over K, then F is algebraic Galois over E, where E is an intermediate field.
...
1
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1answer
43 views
A question related to when is F Galois over K and when F is splitting field
This question was part of my abstract algebra assignment and I was unable to solve 1 part among it. So, I am posting it here as I need help.
Suppose [F : K] is finite. Then the following conditions ...
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0answers
28 views
A question based on intermediate fields and separability
This problem was asked in my Abstract algebra assignment and I was unable to solve it so I am asking for help here.
Let E be an intermediate field .
(a) If u $\in$ F is separable over K, then u is ...
1
vote
1answer
43 views
Let $K \subset L$ be a finite extension with $p = \text{char}(K) > 0.$ Prove $[L : K]_s \cdot p^k= [L : K]$ for $k \in \mathbb{Z}_{\geq 0}$.
This was part of a two part question which first asked to show that $[L : K]_s= [K_s: K],$ where $K_s$ is the separable closure of $K$ in $L$. I use this result in my answer. I'm looking for help ...
0
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1answer
23 views
Given a field $K$ of characteristic $p>0$ and $f \in K[x]$ an irreducible polynomial, find the separable closure of $K$ in $L=K[x]/(f).$
I have shown that we can find some irreducible, separable polynomial $g \in K[x]$ such that $f(x)=g(x^{p^n})$ for some $n \in \mathbb{Z}_{\geq 0}.$ What I have so far towards the problem in the title ...
3
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0answers
87 views
On finite separable morphisms
Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective varieties over an algebraically closed field $k$. Let $d$ be the degree of $f$. We have an exact sequence
$$0\...
1
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1answer
111 views
Solvability by radicals of a separable field of characteristic $p$
Preliminary Definitions
A field $K$ is said to be solvable by proper radicals over $F$ if there exist fields $F_0,F_1,\dots,F_m$ such that $F=F_0\subseteq F_1\subseteq F_2\subseteq\dots\subseteq F_m$ ...
1
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2answers
146 views
If $L/K$ is a finite field extension and $L$ is perfect, then $K$ is perfect.
I'm aware that this question has been answered here. However, I wanna try and prove this directly:
Let $L/K$ be a finite field extension, suppose $L$ is perfect and $\text{char}(K) = p > 0$. Show ...
5
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2answers
156 views
tower of separable extension
Let $K⊂L⊂M$ be a tower of fields.
Let $L/K$ and $M/L$ be separable, is it true that $M/K$ is separable?
I guess there are counterexamples, but I cannot point out them.
Thank you for your kind help.[I ...
1
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1answer
30 views
Simple extension of purely inseparable extension
It is Albert "Modern Higher Algebra", chapter 7, section 9, exercises 5 and 6.
Let $K$ be a field of degree $n$ over $F$ of characteristic $p$ such that every quantity $\alpha$ of $K$ is a ...
2
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1answer
38 views
Is $F$ over $\mathbb{F}_2(x)$ separable?
I want to determine whether the extension $F$ over $\mathbb{F}_2(x)$ is separable, where:
$$F=\mathbb{F}_2(x)[t]/ \langle t^2+t+1 \rangle \quad \mbox{and} \quad \mathbb{F}_2 \mbox{ is the finite group ...
0
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0answers
73 views
If char K=0 , then every irreducible polynomial is separable [duplicate]
The following statement was left as exercise in my Field Theory class.
Consider a field K and f $\in K[x] $ an irreducible polynomial. f is separable in some splitting field of f over K if every root ...
0
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0answers
51 views
Show that there exists an integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$.
Let $F$ be a subfield of $\mathbb{C}$ and $a,b \in \mathbb{C}$ be algebraic elements over $F$. Show that there exists an integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$.
We ...
0
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0answers
32 views
The splitting field of a family of separable polynomials is Galois?
Let $I$ be a non empty set and $K$ a field. Let $\phi_i=\{f_i(x)\}_{i \in I}$ be a family of separable polynomials. Now, take $L:K$ be a extension given by $K$ and the roots of all polynomials in $\...
1
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1answer
46 views
Question related to separable extension.
I want to show:
Let $F$ be a field with characteristic 2. If $E/F$ is a separable extension with degree 2, then $E= F(\alpha)$ for some $\alpha\in F$ such that $\alpha^2+\alpha\in F$.
My attempt: As ...
1
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1answer
76 views
Galois action on a tensor product of fields
Let $K$ be a field, $F$ a finite field extension of $K$ and let $L$ be an algebraic closure of $K$. Let $G_K:=\operatorname{Gal}(K^{\text{sep}}|K)$ be the absolute Galois group of $K$. $G_K$ acts on $...
2
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0answers
61 views
Example of non-flat $R \subseteq R[w]$
Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \...
1
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0answers
66 views
Certain separable ring extensions
Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra which is an integral domain having field of fractions $Q(R)$ and $a$ an algebraic element over $R$ ($a$ belongs to some ...
