# Questions tagged [separable-extension]

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74 questions
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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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### $F/K$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Let $F\subseteq L\subseteq K$ be fields such that $K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal.

While studying Patrick Morandi's book "Field and Galois Theory", on page49, I came across the following question: Let $F\subseteq L\subseteq K$ be fields such that $K/L$ is normal and $L/F$ is purely ...
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### 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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### Example for non separable elements?

We say a polynomial $P \in \mathbb{K}[X]$ is seperable (where $\mathbb{K}$ is a field) if and only if $P$ has only simple roots in the algebraic closure of $K$. We say an element $x$ is seperable if ...
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### 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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### Let $D'$ the subring of $D$-integral elements of $E$. Why $D'$ is Dedekind?

I am studying the theorem: Let $D$ be a Dedekind domain, $F$ its field of fractions, $E$ a finite dimensional extension field of $E$, $D'$ the subring of $D$-integral elements of $E$. Then $D'$ is ...
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### 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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### 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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### 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 ...
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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 ...
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 $... 0answers 7 views ###$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$is separable. If not true then give me any counter example. Answer: In ... 0answers 65 views ### Let$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) $is ... 1answer 38 views ### Question about separable polynomial - proof verification Let$K/F $be a separable extension with char$F =p > 0$. Prove that for any given$\alpha \in K , \alpha \in F (K^p) $. I am not sure how to approach this problem. I think Viete's formula is too ... 0answers 44 views ###$\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$is a separable extension. I have currently the ... 0answers 94 views ### Simple extension is splitting field of$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$.... 2answers 46 views ### Is$\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) \...
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 ...