Questions tagged [separable-extension]

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26 questions
512 views

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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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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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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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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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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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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$L_1L_2/K$ 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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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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$\mathbb F_p(t) \leq L$ 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 ...
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 $f$....