# Questions tagged [separable-extension]

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### Why are these statements about separability equivalent?

I am reading something about separable extensions and passed by the following definitions: (Separable degree) Let $E$ be an algebraic extension of a field $F$. and let $\sigma :F\rightarrow L$ be an ...
• 1,029
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### Proof of Theorem 4.6 (Primitive Element Theorem) in Lang's Algebra

The following shows how Lang proved a statement as a part of the primitive element theorem, which asserts that if a finite extension $E$ of a field $k$ is separable over $k$, then there exists an ...
• 131
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### seperability field extensions transitive [duplicate]

in our lecture course we discussed that for finite field extensions the following equivalence holds: Let $K \subseteq M \subseteq L$ finite field extensions. $L / K$ separable $\Leftrightarrow L/M$ ...
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### Separable and irreducible polynomial over $F$

Let $f(x) \in F[X]$ a polynomial with degree 5 and Galois Group $S_5$. Show that $f(x)$ is separable and irreducible over $F$. For irreducibility, I tried by contradiction $$f(x) = g(x) * h(x)$$ ...
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• 11
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### Extension degree must be power of prime [closed]

Problem statement: If $K/F$ is a finite separable extension, and for any field extension $M/F$, $[M:F]$ is divisible by a fixed prime $p$, show that $[K:F]$ is a power of $p$. Primitive element ...
• 369
1 vote
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### Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$.

Question: Let $K=F(u)$ be a separable extension of $F$ with $u^m\in F$ for some positive integer $m$. Show that if the characteristic of $F$ is $p$ and $m=p^tr$, then $u^r\in F$. (I think we also ...
• 2,348
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### Is an algebraic extension of a separably closed field separably closed?

In the book "Rational points on varieties" by Bjorn Poonen, exercise 1.1 (a) states: "Prove that an algebraic extension of a separably closed field is separably closed." Is this ...
1 vote
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• 371
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### Inseparable field extension of degree 2

I have searched for an example of a degree 2 field extension that is not separable. The example I see is the extension $L/K$ where $L=\mathbb{F}_2(\sqrt t), \ K=\mathbb{F}_2(t)$ where $t$ is not a ...
• 1,530
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### Unseparable element becomes separable

Let $L/K$ be an extension of fields. Denote by $p$ the characteristic of $K$. One assumes that $p$ is a a prime. Let $a\in L$ not separable over $K$. Does it exist an integer $s$ such $a^{p^s}$ is ...
• 937
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### Normal closure as the compositum of conjugates

An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242. Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
• 1,137
1 vote
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### Theorem 17.22 from Ian Stewart's Galois Theory

Apply the Frobenius map to minimal polynomials to see that $$[K(\alpha^p+\beta^p):K(\alpha^p+\beta^p,\beta^p)]\leq [K(\alpha+\beta):K(\alpha+\beta,\beta)]$$ and  [K(\alpha^p+\beta^p):K] \leq [K(\...
• 1,137
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### For which $p$ is $\mathbb F_p(X)$ separable over $\mathbb F_p(X^6)$?

Here's what I've got, but I'm not confident it's airtight... Let $K=\mathbb F_p(X^6)$ then $\mathbb F_p(X)=K(X)$ and the polynomial $F(T) = T^6-X^6\in K[T]$ has $X$ as a root. It's irreducible by ...
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### $f(x)=x^4+x+1$ is separable over $\Bbb F_p$ iff $p=229$

I was asked to prove this in my homework, but when I tried looking at the roots of this polynomial over $\Bbb F_{229}$ I noticed $75$ is a double root of this polynomial and $f=(x-75)^2(x^2+150x+158)$....
• 722
1 vote
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### $E$ is the splitting field over $F$ for some separable $f(X) \in F[X]$ implies that $f(X)$ is separable over $\Phi(\Gamma(E/F))$?

I'm going over the proof of Theorem 25.1 in Abstract Algebra, and I'm trying to understand a particular detail concerning separable polynomials. Here's the relevant portion of the theorem statement: ...
• 816
1 vote
39 views

### Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
• 115
1 vote
49 views

### Properties of $k[x(x-1)]_{\langle x(x-1) \rangle} \subseteq k[x]_{\langle x \rangle}$

Let $k$ be aa arbitrary field. Let $R=k[x(x-1)]_{\langle x(x-1) \rangle}$ and let $S=k[x]_{\langle x \rangle}$, $m=x(x-1)R$, $n=xS$, $k(m)=R/m$, $k(n)=S/n$. We have, $mS = n$ (since $x-1$ is ...
• 5,997
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### Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?

I am studying these notes and I am trying to generalize a bit the setting of the Section 3, because there doesn’t seem to be a fundamental reason to only study $p$-adic fields. So all the fields ...
• 27.1k
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### Separability degree of extension as number of intermediate separable extensions.

I read in another SE post that we can think of separability degree in this way but I can't explain why. Specifically, given an algebraic extension L/K, I want to understand this to see why the ...
• 455
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### Ring $R'$ with $R\subset R'\subset T$ where $R$, $T$ integrally closed and obtained by a purely inseparable extension

Let $R$ be an integrally closed domain with field of fractions $K$. Let $M/K$ be a purely inseparable extension and let $T$ be $R$'s integral closure in $M$. Consider a subring $R\subset R'\subset T$ ...
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