Questions tagged [normal-extension]
For questions about normal field extensions.
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If $F/K$ is normal extension and $f \in K[x]$ irreducible and $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ in $F[x]$ then $m_{i}=m_{j}$ for all $i,j$
So I have that question:
Let $F/K$ be a normal extension and $f$ irreducible polynomial in $K[x]$ assume that $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ where $g_i$ is irreducible in $F[x]$ ($m_i \geq 1$) then ...
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A criterion for proving that an algebraic extension $E \subset F$ of fields is normal.
Let $K \subset F$ be fields such that $F$ is an algebraic extension of $K$, if for all elements $\alpha \in F$ there is a field $K \subset E \subset F$ such that $\alpha \in E$ and $E$ is normal over $...
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How to show that the field $\mathbb{F}_2(t^{\frac{1}{3}})$ does not contain a third root of unity
I want to find a non-normal extension in characteristic p. The following extension is not normal :
$$\mathbb{F}_2(t)\subset \mathbb{F}_2(t^{\frac{1}{3}})$$
It's minimal polynomial is : $X^3-t$ and it'...
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Proof of equivalent definitions of normal extensions
Let $F \supset k$ be an algebraic extension.
We have these two characterizations of a normal extension:
a) For every $k$-homomorphism $\sigma: F \to \overline{k}$, we have $\sigma(F) \subset F$
b) If ...
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When normal extensions are normal
I am wondering if the following statement is correct for each of the following properties $\mathcal P:$ normal, separable, and Galois or not:
If $K \subseteq L \subseteq M$ are fields and $M/K$ has ...
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$E/F$ is normal iff $E$ is a splitting field of some $f(x)\in F[x]$, is it always valid?
This result is proven and well known for finite field extensions, however, consider the question:
Let $F$ be a field and let $E/F$ be a finite extension. Suppose that $\alpha_1, \dots , \alpha_k \in E$...
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Every field extension generated by elements of degree two is normal.
Today I found an exercise that asked to demonstrate that every field $F/K$ extension generated by elements of degree 2 is normal.
If the extension were finitely generated, let's say $F=K(\alpha_1,\...
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Profinite topology on a Galois group can also be induced by normal finite intermediate extensions.
This is about an exercise on Fourier Analysis on Number Fields by Dinakar Ramakrishnan & Robert J. Valenza, exercise 1.14(a.i).
Let $K/F$ be a Galois extension with Galois group $G$.
Let $L$ be ...
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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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Does $[KL:L]=[K:K\cap L]$ when $L/(K\cap L)$ is normal?
Let $K$ and $L$ be fields (inside a common ambient field) with $L/(K\cap L)$ normal. Is it always true that $[KL:L]=[K:K\cap L]$?
This is true when $K/(K\cap L)$ is a finite Galois extension, but I am ...
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Finite normal and separable extension
Let $K/k$ be a field extension of degree $n.$
If $K/k$ is separable. Then $K\otimes_{k} K \cong K^n \Leftrightarrow K/k$ is a normal extension.
I have a solution for $\Leftarrow$ direction:
By ...
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For field extension $M/L/K$ with $M/K$ normal, if $\sigma\in\mathrm{Aut}(M/K)$ implies $\sigma(L)\subset M^{\mathrm{Aut}(M/L)}$, must $L/K$ be normal?
Let $M/K$ be a normal extension, $L$ be an intermediate field. Suppose that for every $\sigma\in\mathrm{Aut}(M/K)$, $\sigma(L)\subset M^{\mathrm{Aut}(M/L)}$ ($M^{\mathrm{Aut}(M/L)}$ is the subfield of ...
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Tietze Extension Theorem - Munkres Section 35, Theorem 35.1
I'm reading the proof for Tietze Extension Theorem from Munkres 2nd. Edition, Section 35, it's $\textbf{Theorem 35.1}$. Part $(a)$ is ok but i'm having trouble understanding the second implication, $(...
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Why isn’t $\mathbb{Q}[x]/\langle x^3-2\rangle$ a normal extension over $\mathbb{Q}$?
The extension $\mathbb{Q}[x]/\langle x^3-2\rangle$ is given on my lecture notes as an example of a non normal extension over $\mathbb{Q}$. I understand why $\mathbb{Q}[\sqrt[3]{2}]$ is not: because it ...
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Show there is a tower of cyclic field extensions of prime degree from $\mathbb{Q}(\sqrt[12]{5})$ to $\mathbb{Q}$
Of course, it suffices to find tower of Galois extensions of prime degree, as these would have to be cyclic. My first thought was to try extending $\mathbb{Q}$ first by $\sqrt 5$, then $\mathbb{Q}(\...
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