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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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Proposition 12 Corollary 1, Section 5.6 of Hungerford’s Algebra

Lemma 6.11. Let $F$ be an extension field of $E$, $E$ an extenion field of $K$ and $N$ a normal extension field of $K$ containing $F$. If $r$ is the cardinal number of distinct $E$-monomorphisms $F\to ...
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Proposition 12, Section 5.6 of Hungerford’s Algebra

Let $F$ be a finite dimensional extension field of $K$ and $N$ a normal extension field of $K$ containing $F$. The number of distinct $K$-monomorphisms $F\to N$ is precisely $[F : K]_s$, the separable ...
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Help me to verify the proof of this theorem, which is proving $k$ª $=$ $\prod_{i=1} ^{\infty}$ $\mathbf K_{p_i}$ by using maximality and minimality

I want to verify my proof is true or false. The exercise what I want to prove is under theorem. $\mathbf {Exercise}$: Let $\mathbf k$ be some perfect field and $\mathbf K_{p_i}$ is a compositum of ...
Snailman's user avatar
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The Frobenius Endomorphism is Surjective iff the field is perfect

I'm taking a Galois theory class right now. I've read and understood the proof that the Frobenius endomorphism is surjective iff the field is perfect (working in characteristic $p$). But it just feels ...
Boran Erol's user avatar
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Extension of the base field of an irreducible representation of a finite group stays completely reducible

I was reading chapter $9$ of "Character theory of finite groups" by Isaacs in which he explores the theory of representation of finite groups over arbitrary fields. In his theorem $(9.2)$, ...
GC.'s user avatar
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Characterization of primitive element.

I would like to characterize primitive elements of field extensions. I know the classical characterization that an element $\alpha$ is primitive in a finite Galois extension if and only if all its ...
IAG's user avatar
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Do we always have $|\mathrm{Mor}_K(K(\alpha),F)|$ divide $|\mathrm{Mor}_K(K(\alpha),\overline{K})|$ with $\alpha\in F$ algebraic over $K$?

Let $F/K$ be an algebraic field extension and $\alpha\in F$. Let $m(x)$ be the minimal polynomial of $\alpha$ over $K$. Then $|\mathrm{Mor}_K(K(\alpha),F)|$ is the size of the roots of $m(x)$ in $F$ ...
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Why does an intermediate field $L$ lie between its maximal separable subfield $L_{0}$ and $L_{0}(v)$?

This question is from the hint of the exercise 7, p59 in Kaplansky's book 'Fields and Rings'. Let $M = K(u, v)$ wherer $u$ and $v$ are algebraic over $K$ and $u$ is separable, then $M$ is a simple ...
Shotaro Hidari's user avatar
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Is the class of separable extensions distinguished?

We know, thanks to embeddings, that the class of separable extensions verified the property of the fields tower. That is, in a fields tower $K\subseteq F\subseteq E$, it is true that: $E/K$ is ...
IAG's user avatar
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If $\alpha$ is separable over $K$, then $K(\alpha)/K $ is a separable extension.

There is another post with this question which ask for a proof not using embeddings: If $\alpha$ separable over $F$ then $F(\alpha )/F$ is a separable extension.. I would like just to know the ...
IAG's user avatar
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Function field over a perfect field can be generated by two elements

I have two questions about the following theorem: Theorem: Let $K$ be a perfect field, $F$ a function field in one variable over $K$ (i.e., a finite algebraic extension of $K(t)$). Then there is $x \...
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Proof of a telescoping formula for separable degrees in Hungerford's Algebra.

A corollary on page 287 of Hungerford's Algebra is Corollary 6.13. If $F$ is an extension field of $E$ and $E$ is an extension field of $K$, then $$[F:E]_s[E:K]_s=[F:K]_s\mbox{ and } [F:E]_i[E:K]_i=[...
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Equivalence between definitions of purely inseparable extension

I was doing some algebra today and I came across the term 'purely inseparable extension'. However, I came across two different definitions of this term. We consider algebraic extensions and write $p$ =...
Lucius Aelius Seianus's user avatar
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Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$.

Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$. I am not sure what to do here. I know that if $f(X) = aX^3 + bX^2 + cX + d \in \mathbb{F}_2[X]...
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Showing that any two separable closures of a field are $K$-isomorphic

We call a field $F$ called separably closed if the only separable algebraic extension $F\subset E$ is the trivial extension, that is $E=F$. A separable closure of a field $K$ is a separable algebraic ...
Thora N's user avatar
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Do we know all non-perfect fields? [duplicate]

Except the (rather famous) example $\mathbb F_p(t)= \{ \frac{f(t)}{g(t)}:\ f,g \in \mathbb F_p[t],\ g\neq 0 \}$, which has the inseparable extension containing the one multiple root of $x^p- t$. Do we ...
NotaChoice's user avatar
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$L/K$ be a field extension with $Char(K) = p > 0$ and $[L : K] = n$ cannot be divided by $p$. Show that $L/K$ is separable.

Hey I want to check my solutions for this exercise: Let $K$ be a field with $Char(K) = p > 0$ and let $L/K$ be an extension whose degree $[L : K] = n$ cannot be divided by $p$. Show that $L/K$ is ...
Marco Di Giacomo's user avatar
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Equivalent definition of a perfect field

Let $K$ be a field. I would like to prove that any algebraic extension $L$ of $K$ is separable iff ($\textrm{char} K = 0$ or $K = K^p$ in the case that $\textrm{char} K = p > 0$. I have already ...
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Reducing an inseparable polynomial over the same field to a separable polynomial over a field

Description: Let $F$ be a perfect field and $p(x)$ a polynomial over $F$ with multiple roots. Show that there is a polynomial $q(x)$ over $F$ whose distinct roots are the same as the distinct roots of ...
Marcus Camilus's user avatar
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The minimal polynomial splits over a field of prime characteristic?

The statement of the exercise from the textbook goes: Let $a \in E$, where $E$ is an algebraic extension of a field $F $ of prime characteristic $p$. Let $m(X)$ be the minimal polynomial of $a$ over ...
NotaChoice's user avatar
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Separability over a compositum of fields

Suppose $K/E$ and $K/F$ are two separable (nonalgebraic) field extensions. Is $K$ necessarily separable over the compositum $EF$?
revan's user avatar
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Degree of the perfection of a field

I am currently studying perfection in the context of Galois Theory. For a field $K$ of characteristic $p$ and algebraic closure $K'$, we define $$ K^{\text{perf}} = \{ a \in K' : \text{there exists } ...
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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 ...
rulerandcompass's user avatar
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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 $X$....
Analyse300's user avatar
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2 answers
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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 ...
unoriginalname's user avatar
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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 ...
Jpj's user avatar
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1 answer
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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 $f$ ...
Margaret's user avatar
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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 ...
Waaal's user avatar
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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 ...
Luigi Traino's user avatar
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1 answer
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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 ...
Amanda Wealth's user avatar
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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 $\hat{\phi}$ ...
user631874's user avatar
1 vote
3 answers
90 views

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 \...
Matsmir's user avatar
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2 votes
0 answers
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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,...
Ariel Yael's user avatar
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1 answer
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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\...
SalutaFungo's user avatar
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22 views

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 ...
Ariel Yael's user avatar
2 votes
0 answers
44 views

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\...
H.Yang's user avatar
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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 ...
NiRvanA's user avatar
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1 answer
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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 ...
Brais Romero's user avatar
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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 $k\geq 1$...
user1104937's user avatar
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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 $L$,...
wanymose's user avatar
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1 answer
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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 ...
Kieran McShane's user avatar
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 ...
Albert's user avatar
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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 ...
wanymose's user avatar
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2 votes
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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)= ...
cut's user avatar
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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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