Questions tagged [separable-extension]
The separable-extension tag has no usage guidance.
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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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Are all compact sets in a separable locally compact space G_delta sets?
In Halmos' Measure theory Theorem E from $\S$50 states that a compact set $C$ in a separable locally compact space $X$ is a $G_\delta$ set. The proof goes by the following logic: for $x \in X \...
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Show $\mathbb{Q}(\sqrt{3+\sqrt{3}})/\mathbb{Q}$ is not a normal extension and find it's normal closure [duplicate]
I want to show $\mathbb{Q}(\sqrt{3+\sqrt{3}})/\mathbb{Q}$ is not a normal extension and conclude that the normal closure is $\mathbb{Q}(\sqrt{3+\sqrt{3}},\sqrt{3-\sqrt{3}})$. After knowing the former,...
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Proof that on infinite fields, finite separable extensions are all simple.
Question:
Let $k$ be a infinite field. If $F = k(\alpha_1,...,\alpha_r)$, with each $\alpha_i$ separable over $k$, prove that there exist $c_1,...,c_r \in k$ such that $F = k(c_1\alpha_1+...+c_r\...
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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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Classification of separable rings
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\...
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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 ...
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If $I=F$ , $K/S$ purely inseparable, $S/F$ separable then $K/I$ is not separable
In Patrick Morandi's book Field and Galois Theory Example 4.24 they write Let $k$ be a field of characteristic 2, let $F=k(x, y)$ and $S=F(u)$, where $u$ is a root of $t^2+t+x$, and let $K=S(\sqrt{u y}...
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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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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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Separable extensions and simple roots
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 ...
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How to show that the image of the morphism $X \mapsto X^{p^n}$ equals the separable closure.
Let $K$ be a field with $\operatorname{char}(K) = p > 0$ and let $L = K[X]/(f)$ for some irreducible polynomial $f \in K[X]$. We know that there is a separable polynomial $q_{sep}\in K[X]$ such ...
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Finite extension of a characteristic p field is separable if and only if $E=F(E^p)$
I am trying to show that if $E$ is an extension of a field $F$ with char$(F)$=p, prime, then the extension is separable if and only if $E=F(E^p).$ I have proven that if $\{e_1,\ldots ,e_n\}$ is a ...
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Separable polynomial in an algebraic closure
There is a separable polynomial $G(X)$ over a field $K$.
There is another polynomial $F(X)$ over the same field $K$ such that all roots of $G(X)$ (in an algebraic closure of $K$) are also roots of $F(...
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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 ...
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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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field extension is simple
I am working on the following problem:
Let $L/K$ be a field extension of degree $6$. Show that $L/K$ has a primitve element.
My idea is to use the theorem of primitive elements. So, if I prove that $L=...
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Question about the minimal polynomial of an inseparable element in a field
Let $K/F$ be a field extension. The characteristic of F is prime $p$. Suppose $\beta \in K$
is an inseparable element of degree $p$, show the minimal polynomial of $\beta$ in $F$.
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Why is a field extension separable if and only if the discriminant of the basis of the field extension is nonzero?
Let $L/K$ be a finite dimensional field extension. We define the trace function $T_{L/K}(x)$ of $L$ over $K$ as $T_{L/K}(x)=\textrm{trace}(r_x)$, where $x\in L$ and $r_x$ is matrix given by the ...
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Composition of two simple field extension with coprime degree is simple?
Let $E/F$ be a simple field extension of degree $m$ and $L/E$ be a simple field extension of degree $n$, where $\gcd(m,n)=1$. Is it necessary that $L/F$ is simple?
In the setting of characterstic $0$ ...
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Proving the Existence of an Intermediate Field with Special Properties in an Inseparable Field Extension
I am reading Algebraic Number Fields by Janusz. In the middle of one of his proofs on page 23, he makes the following claim:
Suppose $K$ is a field and $L$ a finite field extension that is not ...
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Why is $K^{\mathrm{sep}}$ dense in $K_v^{\mathrm{sep}}$?
Let $K$ be a global field and $v$ be a non-archimedean place of $K$. Given a field $k$, let $k^{\mathrm{sep}}$ be a separable closure inside an algebraic closure $\overline k$.
I would like to know ...
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Infinite algebraic field extension of a finite field is normal and separable
I am trying to prove that every infinite algebraic field extension $K$ of a finite field $\mathbb{F}$ is separable and normal. I know how to do it for finite $K$, but I am struggling to see why the ...
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Finding the roots of $x^5-2$. My question is about finding the $5^{th}$ roots of 2.While the associated question is about finding roots of unity [duplicate]
I know the roots of polynomial $x^5-2$ are $\sqrt[5]{2},\zeta_{5}^{i}\sqrt[5]{2}$, where $0<i<5$, but I don’t know how we get these roots except $\sqrt[5]{2}$. Can someone guide me the method of ...
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When is the product of two separable polynomials necessarily separable?
Obviously, this doesn't hold in general. I was watching this video and at 11:40 the creator claims that if $g = fh$ and $g$ is inseparable, while $f$ is separable, then $h$ must not be separable. This ...
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Why do two minimal polynomials of $\alpha$ over different extensions share the multiplicity of $\alpha$?
I am reading this PDF and concerned about Lemma 3. I will copy it here for completeness.
Lemma 3. Assume that $K \subseteq E \subseteq L$ is a tower of finite field extensions. If an element $\alpha$ ...
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Are inseparable elements of an extension field closed under addition?
I know that both the separable and purely inseparable elements form fields. It seems very likely false that the inseparable elements form a field, but in a proof I'm developing the following fact ...
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A field extension is separable iff it is contained in a Galois Extension
The following statement appears in Corollary 4.10 of Morandi's book titled Field and Galois Theory
Let L be a finite extension of F. Then, L is separable over F if and only if it is contained in a ...
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Why is an element separable only if it is of the form $\alpha^p$ for some $\alpha$?
Here is a problem from Allan Clark's Abstract Algebra book:
Let $F$ be a field, $p > 0$ the characteristic of $F$ and $E$ a finite algebraic extension of $F$. Denote by $E^{(p)}$ the minimal ...
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Question about what is a separable polynomial
Here is the definition of separable polynomial as I understand it.
Let $K$ be a splitting field for a polynomial $f(x)\in F[x]$. If $f(x)$ is irreducible, then $f(x)$ is separable if the roots in $K$ ...
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Show $\alpha$ is separable over a field $F$ iff $F(\alpha) = F(\alpha^p)$.
Let $F$ be a field of characteristic $p \neq 0$ and $\alpha$ an element of an extension $E$. Show $\alpha$ is separable over $F$ iff $F(\alpha) = F(\alpha^p)$.
For the backward direction, I haven't ...
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Degree of a separable extension of a Dedekind ring's quotient field
I'm studying "Algebraic number theory" written by S. Lang. In proposition 21 of chapter 1.7 it says if $A$ is a Dedekind ring and $K$ its quotient field, for a separable extension $L/K$, $[L:...
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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 ...
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$K(u, v) = K(u+v)$ 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 ...
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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 ...
