Questions tagged [separable-extension]

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Is the following statement about splitting fields true?

Let us take a look at the following definitions: Let $F$ be a field and $f(x)\in F[x]$ then a field extension $E$ of $F$ is said to be the splitting field of $f$ over $F$ if $f$ splits completely in $...
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tensor with the separable closure

Let $l/k$ be a finite field extension and $K$ the separable closure of $k$. The finite $K$-algebra $l \otimes _k K$ is a product of finitely many separable fields $L_i$ over $l$. Are these fields $L_i$...
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$K(u, v) = K(u+v)$ when characteristic of $K=0$?

Let $F$ be a field extension of $K$ where $u, v$ are in $F$ such that $u$ is separable over $K$ and $v$ is purely inseparable over $K$. The answer to this question shows that $K(u, v) = K(u+v)$ when ...
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Existence of $p^{th}$ root in an inseparable extension

Let $K= \mathbb{F}_p(t)$ i.e. the field of rational functions over $\mathbb{F}_p$. I'm trying to prove that if $x$ is inseparable over $K$, then $K(x)$ contains a $p^{th}$ root of $t$. I tried to find ...
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How do I prove the following statement about separable degrees in towers.

I have the following problem: Let $k\subset F\subset E$ be a tower, then $$[E:k]_S=[E:F]_S\cdot [F:k]_S$$ I have given the following definition of the separable degree Let $E$ be an algebraic ...
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Chapter 1, Proposition 14 of Lang's number theory - further explanation for normality

Attached is prop 14 on page 15 of Lang's 'algebraic number theory' 2ed. Note, $G_\mathfrak{B} \subset G$ is the subgroup fixing $\mathfrak{B}$ Can someone perhaps add a sentence or two to explain ...
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Is there a nontrivial but short example where the separable degree is strictly less than the degree of a field extension?

I consider the following theorem: Let $E\supset F\supset k$ be a tower. Then $$[E:k]_s=[E:F]_s\cdot [F:k]_s.$$ Furthermore if $E$ is finite over $k$ then $[E:k]_s$ is finite and $$[E:k]_s\leq [E:k].$$ ...
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Why is this statement about separable degree true?

I have the following problem: Let $E$ be separable over $k$. Consider the tower $$k\subset k(\alpha_1)\subset...\subset k(\alpha_1,...,\alpha_n)$$Then since each $\alpha_i$ is separable over $k$ each ...
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Is there a way to prove theorem $4.1$ about separable degree of S.Lang on page $240$ without using proposition $2.7$?

I'm looking at theorem $4.1$ about separable degree of S.Lang on page $240$. Let $E\supset F\supset k$ be a tower, then $$[E:k]_S=[E:F]_S\cdot [F:k]_S$$Furthermore if $E$ is finite over $k$ then $[E:...
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Why are these statements about separability equivalent?

I am reading something about separable extensions and passed by the following definitions: (Separable degree) Let $E$ be an algebraic extension of a field $F$. and let $\sigma :F\rightarrow L$ be an ...
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Proof of Theorem 4.6 (Primitive Element Theorem) in Lang's Algebra

The following shows how Lang proved a statement as a part of the primitive element theorem, which asserts that if a finite extension $E$ of a field $k$ is separable over $k$, then there exists an ...
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seperability field extensions transitive [duplicate]

in our lecture course we discussed that for finite field extensions the following equivalence holds: Let $K \subseteq M \subseteq L$ finite field extensions. $L / K$ separable $\Leftrightarrow L/M$ ...
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Separable and irreducible polynomial over $F$

Let $f(x) \in F[X]$ a polynomial with degree 5 and Galois Group $S_5$. Show that $f(x)$ is separable and irreducible over $F$. For irreducibility, I tried by contradiction $$ f(x) = g(x) * h(x) $$ ...
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Suppose char$(F) = p$ and $E/F$ with $[E:F] = p$. Then if there is *one* inseparable element then every element in $E\setminus F$ is inseparable

Suppose that $F$ is a field of characteristic $p > 0$ and $[E:F] = p$. Then suppose that there is some $\alpha \in E\setminus F$ with an inseparable minimal polynomial over $F$. Then every $\alpha \...
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Separability degree of $\mathbb{Q}(\sqrt{2}, \sqrt[3]{3})/\mathbb{Q}$

I want to find $[L=\mathbb{Q}(\sqrt{2}, \sqrt[3]{3}):K=\mathbb{Q}]_s$. In lecture I learned that for every algebraic closure $\iota_a:K\rightarrow K^a$ it's true that... \begin{align*} [L:K]_s = \#\{\...
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How to obtain an expression for a complex differential equation

Let us assume that we have the following expression: $\frac{dx'(t,θ)}{dt} = Ax'(t,θ) + B(t)$ where $x'(t,θ) = \frac{\partial x(t,θ)}{\partial θ}$ in which $A$ is constant and $B(t)$ is a function of $...
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Extension degree must be power of prime [closed]

Problem statement: If $K/F$ is a finite separable extension, and for any field extension $M/F$, $[M:F]$ is divisible by a fixed prime $p$, show that $[K:F]$ is a power of $p$. Primitive element ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 $a\in E$. ...
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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 $...
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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 ...
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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 ...
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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 $E^\text{a}$) ...
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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(\...
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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 ...
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3 votes
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$f(x)=x^4+x+1$ 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 $f=(x-75)^2(x^2+150x+158)$....
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$E$ 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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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 $\overline k$ ...
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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 ...
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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 ...
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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 ...
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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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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]...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 $a\in K$...
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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 ...
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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. ...
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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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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 ...
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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 ...
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