# Questions tagged [separable-extension]

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### Is there a separable extension of degree 21?

Given the field $F= \mathbb{F}_3$ and a transcendental $t$, I am trying to find an intermediate field $$F(t^{1/63}) \supset E \supset F(t)$$ where $[F(t^{1/63}): E]= 21$ and the extension is ...
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### Purely inseparable field extensions - proving $\alpha^{p^m} \in F$ implies $m_\alpha = x^{p^m}-a^{p^m}$

I'm reading Isaacs' "Algebra: A Graduate Course" and I don't really understand the proof for the implication (2) $\Rightarrow$ (3) in Theorem 19.10 (page 298): Suppose $F\subseteq E$ is an ...
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### Separable degree of an intermediate extension in a normal extension

I am currently studying the separable degree of an algebraic extension. I have come across the following result: if $K \subset L \subset M$ are algebraic extensions and $M$ is normal over $K$, show ...
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### Action of some Galois group on scheme of finite type

Let $X$ be a $k$ variety (i.e. a $k-$scheme of finite type). Let $k^{s} \subset \bar{k}$ be the separable closure of $k$. I will wright $X(k^{s})$ for the set of $k$ morphism from $Spec(k^{s})$ to $X$....
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### What are the advantages of separable extensions?

I understand the definition of separable extensions very well. But I want to understand whether it holds importance as an individual concept, or does it only make sense when it's paired along with the ...
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### Finite separable field extension of a non-perfect field $K$ of characteristic $p > 0$ that has degree divisible by $p$.

I'm asked to find a finite separable field extension of a non-perfect field $K$ of characteristic $p > 0$ that has degree divisible by $p$, but I don't see the solution. Since $K$ is not perfect I ...
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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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Let $C$ be a symmetric monoidal category. A unital associative algebra $(A,m:A\otimes A\to A)$ is called separable if there exists an $A$-$A$-bimodule homomorphism $d: A\to A\otimes A$ such that $m\... • 61 1 vote 1 answer 88 views ### Is$x^{100} - x^2 + 1$separable in an algebraic closure of$\mathbb{F}_2$My approach:$f'(X) = 100x^{99} - 2x = 0x^{99} - 0x = 0$since in$\mathbb{F}_2$. So the$\gcd(f,f') = f > 1$, thus not separable. On the other hand,$f(0) \neq 0 \neq f(1)$, so irreducible. But ... • 57 0 votes 1 answer 94 views ### 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 ... • 119 0 votes 1 answer 71 views ### 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$k\geq 1$... • 153 0 votes 0 answers 79 views ### 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$L$,... • 572 2 votes 1 answer 54 views ### 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 ... 3 votes 1 answer 137 views ### 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 ... • 3,052 0 votes 1 answer 46 views ### 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 ... • 572 2 votes 1 answer 101 views ### 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$a \in Kp:=x^n-x-a \in K[x]$Show that$p$is separable Show that if$p$has a root in$K$, then$p$splits into linear factors in$K[x]$... • 572 0 votes 1 answer 64 views ### Minimum polynomial and inseparable degree Let$\alpha$be algebraic over$K$. Prove: $$f_K ^{\alpha} =\prod_{\sigma\in X(K(\alpha)/K)} (X-\sigma(\alpha))^i$$ Where$i$is the inseparable degree of$K(\alpha)$over$K$. Also,$X(K(\alpha)/K)= ...
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An element $\alpha\in \mathbb F$ is algebraic w.r.t a field extension $\mathbb E\subset \mathbb F$ if it's the root of some polynomial over $\mathbb E$. One can prove the sum, product, and inverse of ...