# Questions tagged [separable-extension]

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### Semisimplicity implies separability for a perfect field

Let $k$ be a field, $A$ a finite-dimensional semisimple $k$-algebra. If $k$ is a perfect field (every finite field extension of $k$ is seperable), then $A$ is separable. I know a proof that uses the ...
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### When are coordinate rings separable?

Let $k$ be a field, and $f\in k[x]$ be a polynomial. Consider the coordinate ring $k[x]/(f)$. This is a $k$-algebra. I have seen people using the statement that this $k$-algebra is separable iff $f$ ...
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### Splitting field of a separable (and irreducible) polynomial is separable

I was struggling proving this and didn't manage to find any solution here that felt understandable for my level, so I am submitting my best idea, which seems right. Let $L$ be a splitting field of a ...
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### Is $\alpha \in \mathbb{F}_{p^n}$ separable over $\mathbb{F}_p(t)$?

I know that $\alpha \in \mathbb{F}_{p^n}$ is separable over $\mathbb{F}_p$ because finite fields are perfect. This means that the minimal polynomial $\mu_\alpha(x)\in\mathbb{F}_p[x]$ is separable. Is ...
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### Understanding proof of Proposition 5.49 of the Gortz's Algebraic Geometry book.

I am reading the Gortz's Algebraic Geometry, Proposition 5.49 and stuck at some point. First, I propose a question. Q. Let $Y = \operatorname{Spec}B$ is affine reduced $k$-scheme ( $k$ is a field ). ...
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### Separable field extension with bounded simple extensions

I’d like to get some help about an exercise in Field theory (J. Bastida, ‘Field extensions and Galois theory’, p. 157, problem 2). Let K be a field and let L be a separable extension of K. Suppose ...
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### Corollary 6.4:Arithmetic of Elliptic Curves,Silverman

Notation:$\hat{\phi}$ is the dual isogeny for the Frobenius morphism($\phi$). In proving (c) part of this corollary,we have 2 cases.Either $\hat{\phi}$ is separable or inseparable.Suppose $\hat{\phi}$ ...
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### prove $\deg_{\alpha,L}\mid\deg_{\alpha,K}$ if $\alpha$ is separable and $K(\alpha)/K$ is normal

I'm trying to prove the following: Let $K$ be a field, $\alpha \in \overline{K}$ a separable element s.t. $K(\alpha)/K$ is normal, and let $L/K$ be some finite subextension of $\overline{K}$. Prove ...
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### Are all algebraic extensions of finite fields separable? What about fields of characteristic p in general?

I know that all algebraic extensions of fields of characteristic $0$ are separable, but what about a field of characteristic $p$, for example, $\mathbb{F}_7$? I know that, for a finite field of ...
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### Separable extension characterization

Let $F$ be a field of characteristic $\operatorname{char}F=p\neq 0$. It is well-known that a simple extension $F<F(\alpha)$ is separable if and only if $F(\alpha^{p^k})=F(\alpha)$, for any $k\geq 1$...
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### If $L/K$ is normal extension $\Rightarrow K_s/K$ is normal extension

If is $L/K$ a normal extension, then it follows that $K_s/K$ is a normal extension. Definition of normal extension: Let $L/K$ be an algebraic extension and $\overline{L}$ be a algebraic closure of $L$,...
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### Field extension $L/K$ such that every element has degree $1$ or $2$ over $K$

Let $L$ be a field extension of a field $K$ of characteristic $\neq 2$ such that every element of $L\setminus K$ has degree $2$ over $K$, can we show that $[L:K]=2$ by elementary methods, without ...
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### Example of a field on which every irreducible polynomial has degree a power of $p$

Exercise A-47 in Milne's Fields and Galois Theory notes asks to prove that if $p$ is a prime number and $F$ is a field of characteristic zero such that every irreducible polynomial $f(X)\in F[X]$ has ...
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### Let $L/K$ be finite extension, is $L/K_s$ purley inseperable

I was thinking about the following: Let $L/K$ be a finite extension and $K_s=\{a \in L : a \text{ is separable over } K\}$ Is $L/K_s$ purely inseparable? I will start with the definiton: The ...
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### If $p:=x^n-x-a$ has a root in $K$, then $p$ splits into linear factors in $K[X]$

Let $K$ be a field with $\text{char}K=n>0$. Define for $a \in K$ $p:=x^n-x-a \in K[x]$ Show that $p$ is separable Show that if $p$ has a root in $K$, then $p$ splits into linear factors in $K[x]$ ...
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### Tensor product criteria of an extension separability

Given an extension $L|\mathbb{k}$ I need to prove that the ring $L\otimes_{\mathbb{k}}L$ embeds into a direct product of algebraic extensions of $L$ if and only if $L|\mathbb{k}$ is separable. I can ...
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### Show for algebraic $L/K$ if $L$ perfect, $K$ not necessarily perfect.

I just need a simple example to show the statement, but Im doubting the one I thought of. I have proof that $K$ perfect $\iff L/K$ separable. So to show the statement we need some $L$ that is perfect ...
1 vote
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### Irreductibility of $x^p-t$ for fields $F$ with characteristic $p$ and such that $t$ has not a $p$-root in $F$

I am having problems to understand the the proof of following Lemma 9.14.2 in: https://stacks.math.columbia.edu/tag/09HF Lemma 9.14.2. Let $p$ be a prime number. Let $F$ be a field of characteristic p....
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### Last part of a theorem on automorphisms of $\mathbb{F}_q$ in Lang's algebra

On Lang's Algebra theorem 5.4 states The group of automorphisms of $\mathbb{F}_q$ where $q=p^n$ is cyclic of degree $n$, generated by the Frobenius isomorphism. On the last part of the theorem Lang ...
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### Does anyone know the Reflection method in solving PDEs?

I am looking at solutions for a diffusion equation, where we have: \begin{equation} \frac{1}{4}u_{xx}=u_t , \ \ \ \ \ \ \, 0<x<\infty, t>0 \\ u_x(0,t)=0 \\ u(x,0)=\begin{cases} 6 \ \ \ \ 0 \...
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### Prove that $A$ is a field [duplicate]

I've trying to prove some theorem about Galois-Hopf Theory. It's details are not necessary to understand my question. I want to prove the following: Let $F/K$ be a Galois field extension. Let $A$ be ...
### $K(u, v) = K(u+v)$ 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 ...
### 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 ...