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If $L/K$ is a finite extension of fields, then we have the norm map $N_{L/K}:L \rightarrow K$ given by $N_{L/K}(x)=det(l_x) \text{, } \forall x \in L$, where $l_x:L\rightarrow L$ is the $K-$linear map ...
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Let $F$ be any field and let $E/F$ be a nontrivial field extension. Suppose this field extension has the property that for every $x\in E$, there exists a positive integer $n$ such that $x^n\in F$. ...
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Let $k$ is a field, $A$ and $B$ are finite dimensional algebras over $k$. It is well-known that $gl.\dim(A\otimes_k B)\leq gl.\dim(A)+gl.\dim(B)$ if $k$ is a perfect field, where $gl.\dim(A)$ is the ...
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Let $[E:F]_{\text{s}}$ denote the separable degree of the algebraic extension $E/F$. Let $\bar F$ be a fixed algebraic closure of $F$. In Lang's Algebra(GTM211), the separable degree is defined to be ...
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The following theorem is from Lorenz's "Algebra, Volume II: Fields with Structure, Algebras and Advanced Topics", p. 167, F20, about a characterization of a separable field extension using a ...
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Given the algebraic definition of a $\mathbb{Z}$-linear derivation over a commutative ring $D:R\to R$ with $D(a+b)=D(a)+D(b)$ and $D(a \cdot b)=D(a) \cdot b+a \cdot D(b),$ there is always the trivial ...
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I have the following question which I'm stuck on If $p(x)=x^{16}+x^{8}+x^{4}+x^{2}+1\in \mathbb{Z}_{2}[x]$, is $p(x)$ separable? My work so far is the following: As $p(x)$ is a polynomial in $x^{p}$ ...
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$(\star)$ Let $L = F(x)$ be the field of rational functions over a perfect field $F$ of characteristic $p$ . Let $K = F(f)$ for some rational function $f$ . Prove that $L/L_{T.I}$ is ...
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I am teaching myself algebra using a lecture notes, which has the following statement Let $K$ be a field and $p\in K[x]$ be irreducible. Then $p$ is separable if and only if $p'\neq 0$. I have ...
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Let $E/F$ be an algebraic field extension, denote $E_{sep}$ by the separable closure of $F$ in $E$. We know if $E/F$ is normal then $E_{sep}/F$ is normal, see tag 0EXK. The reverse should be false and ...
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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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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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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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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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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 ...
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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 $(9.2)$, ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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 $p$ =...
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Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$. 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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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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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\...
H.Yang's user avatar
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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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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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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 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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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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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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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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