Questions tagged [separable-extension]

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73 questions
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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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$F/K$ 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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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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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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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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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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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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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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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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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 $K$ 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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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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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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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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$\mathbb F_p(t) \leq L$ 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 ...
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 ...