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Questions tagged [normal-extension]

For questions about normal field extensions.

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Why this extension is Galois?

Taken from Algebra : Chapter 0 by Paolo Aluffi "Let $f(x) \in \mathbb{C}[x]$ be a nonconstant polynomial; we have to prove that $f(x)$ has roots in $\mathbb{C}$. Note that if $f(x)$ has no roots ...
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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 ...
Degenerate D's user avatar
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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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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 ...
Thomas Browning's user avatar
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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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Simple normal extension

I was asked to prove that if $a$ is any root of $13x^{4} - 29x^{2} + 13$ then the extension $\mathbb{Q}(a)/\mathbb{Q}$ is a normal extension. We see that $-a$ is actually another root of the ...
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Is the extension $\mathbb{Q}(\sqrt{7+\sqrt{7}})/\mathbb{Q}$ a Galois extension? [duplicate]

I'm preparing for an algebra exam, and in previous exams I found the following exercise: Let $\alpha = \sqrt{7+\sqrt{7}}$. Show that $\mathbb{Q}(\alpha)$ is a Galois extension of $\mathbb{Q}$ of ...
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Show that there exist $K_0 = \mathbb Q ⊆ K_1 ⊆ K_2 ⊆ K_3 ⊆ K_4 = L$ , where, for $i = 1, . . . , 4$ , $|K_i : K_{i−1}| = 3$.

Let $\mathbb Q ⊆ L$ be fields. Assume that $L$ is a normal extension of $\mathbb Q$ and $|L : \mathbb Q| = 81$. Show that there exist intermediate fields $K_1$, $K_2$, and $K_3$ with $K_0 = \mathbb Q ⊆...
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Root of the polynomial $x^3-x-1$ over $\mathbb{Q}$ does not generate normal extension

Show that the field generated by a root of $f(x)$ where $f(x)=x^3-x-1$ over $\mathbb{Q}$ is not normal over $\mathbb{Q}$. First in order to mention some extension I showed $f(x)=x^3-x-1$ is ...
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Non-normal extension of Q

The counterexamples of normal extensions over $\mathbb{Q}$ that I could find are basically all in the form of $E = \mathbb{Q}(\sqrt[n]{\alpha})$ (with $x^n - \alpha$ being its minimal polynomial; and ...
cake-and-tea's user avatar
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Find a particular intermediate field $M$ such that $\mathbb{Q}\subset M\subset\mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})$.

Problem: Let $\alpha=\sqrt{\frac{3+i\sqrt{7}}{2}}$ and $K=\mathbb{Q}(\alpha)$. Find the fixed field $M=\{x\in \mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})|\sigma(x)=x\}$, where $\sigma$ is the ...
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Splitting fields are normal using symmetric polynomials.

I was told it is possible to prove splitting fields are normal using the Fundamental Theorem of Symmetric Polynomials rather than the usual approach. Does anyone have hints or a reference for this? ...
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What does an embedding over $K$ mean?

I have some understanding problem. In the topic of extensions they always speak about embeddings over some field $K$ and i have absolutly no idea what this means. One example is the following: Let $K$...
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What does it mean that a normal extension remain normal under lifting?

I consider the book "Algebra" by Serge Lang and on page 238 he has the theorem 3.4 saying that normal extensions remain normal under lifting. I don't see what he means by that, and therefore ...
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Can someone explain me what they did in the proof that every extension of degree 2 is normal?

I consider the following proof that every extension of degree $2$ is normal and I got stuck somewhere: Let $K$ be an extension of $F$ with $[K:F]=2$. Then $K=F(\alpha)$ where $\alpha$ is a root of an ...
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Is $F\hat{E_s}/E$ normal, where $F/E$ is a finite field extension?

Let $F/E\,$ be a finite field extension, let $E_s$ be the separable closure of $E$ in $F$ and let $\hat{E_s}$ be its normal closure. Then $\hat{E_s}/E$ is Galois and $F\hat{E_s}/\hat{E_s}$ is purely ...
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On normal extension, a square root of discriminant is in field F?

The probrem I have now is Let $F$ be a Field, and $f(X) = X^3+aX+b \in F[X]$ be a irreducible polynomial. Also, $\alpha, \beta, \gamma \in E$ ($E$ is the smallest splitting field of $f$) are roots of ...
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If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?

Let $k\subseteq F$ denote an algebraic field extension and let $\alpha\in F$ having $f\in k[x]$ as its minimal polynomial. My question: If $\beta\in F$ is a root of $f$ then does there exists some $\...
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If $K \subset L \subset \overline{L}=\overline{K}$, then is it true that $\sigma\in Aut(\overline{L}\mid K)$ satisfies $\sigma|_L\in Aut(L\mid K)$?

We consider a tower of fields $K \subset L \subset \overline{L}=\overline{K}$. Is it true that, given a $\sigma\in \text{Aut}(\overline{L}\mid K)$, we can restrict it to an automorphism of $L$, that ...
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Extended automorphisms

Let $F < E < K$ be a tower of field extensions and assume $K$ is a normal extension over $F$. It's well-known that any embedding in $E$ fixing $F$ can be extended to an isomorphism in $K$. One ...
Hoang Nguyen's user avatar
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galois automorphism and normal (extension) of normal is normal

Problem Statement: Given $F \subseteq L \subseteq K$ field extensions such that $K/L$ and $L/F$ are galois, and any automorphism of $L/F$ extends to an automorphism of $K$, show that $K/F$ is galois. ...
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Compositum of normal extensions

Given two normal (finite) extensions $E_1,E_2$ of $F$, such that $E_1,E_2 \subseteq \bar{F}$, prove that $E_1E_2 = \{\sum_{i=1}^d e_1e_2: e_1 \in E_1, e_2 \in E_2, d>0\}$ is normal. I see the ...
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Prove that $\mathbb{Q}(\sqrt{2+\sqrt{2}}) / \mathbb{Q}$ is a normal extension

I need to prove that $\mathbb{Q}(\sqrt{2+\sqrt{2}}) / \mathbb{Q}$ is a normal extension and $\mathbb{Q}(\sqrt{2+\sqrt{2}}, i)=\mathbb{Q}(\phi)$, where $\phi^{4}=i$. For the first part i use the ...
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$L$ be an algebraically closed field and $\sigma \in \operatorname{Aut}(L)$, if $F=L^{\sigma}$, then every finite extension $E/F$ is cyclic.

Let $L$ be an algebraically closed field and $\sigma \in \operatorname{Aut}(L)$. Let $F$ be the fixed field of $\sigma$, i.e., $F=\{x\in L:\sigma(x)=x\}$. Then I have to show that every finite ...
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Splitting Field of $x^6-3x^4+3x^2-3$ \ Normal Extension \ Subfields

i'm trying to solve the problem above. Let's name L the splitting field. Using $x^2=t$, i have found 6 roots $$\alpha=\sqrt{1+\sqrt[3]{2}} \\ \beta=\sqrt{1+\varepsilon\sqrt[3]{2}} \\ \gamma=\sqrt{1+\...
TyrannosardusRex's user avatar
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Irreducible polynomial over normal extension

Let $E/F$ be a normal extension and a irreducible polynomial $f(X)∈F[X]$. Prove that all irreducible factors of $f(x)$ in $E[X]$ have the same degree. More explicit: if $f$ factors as the product of ...
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Prove that $\Bbb{Q}(\sqrt2,\sqrt3,u)|_\Bbb{Q}$ is normal where $u^2=(9-5\sqrt3)(2-\sqrt2)$

As $u$ satisfies the polynomial $X^2-(9-5\sqrt3)(2-\sqrt2)$ over $\Bbb{Q}(\sqrt2,\sqrt3)$ and it is irreducible over $\Bbb{Q}(\sqrt2,\sqrt3)$. We have $[\Bbb{Q}(\sqrt2,\sqrt3,u):\Bbb{Q}(\sqrt2,\sqrt3)]...
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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 $E^\text{a}$) ...
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Find normal closure of $\mathbb Q\left( (1+i)\sqrt[4]{5} \right)/\mathbb Q$

I've been solving problems from my Galois Theory course, and I've had problems finishing this one. It says: Being $a=(1+i)\sqrt[4] 5$, find the normal closures of these field extensions: $\mathbb Q(...
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A problem on finite normal extension .

$\mathbf {The \ Problem \ is}:$ Question number $11.$ $\mathbf {My \ approach}:$ Actually, we know $L$ being a finite normal extension, is a splitting field of some polynomial $p(x)$ over $k.$ A hint ...
Rabi Kumar Chakraborty's user avatar
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Understanding the definition of normal field extension

We let $E/F$ be an algebraic extension of a field $F$, and define $S$ to be the set of all $F$-conjugates of the elements of $E$. Now we define $F(S)$ the be the field generated by $S$ over $F$. rove: ...
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$F_1F_2$ Normal extension of k

Let $F_1$ and $F_2$ be finite normal extensions of $k$ contained in the field $E$. Prove that their composite $F_1F_2$ (the smallest subfield of $E$ containing both $F_1$ and $F_2$) is also a normal ...
user867777's user avatar
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Checking if an Extension is normal

I want to see if my reasoning is correct. I am given a list o field extensions and asked to determine which ones are normal extensions: (a) $\mathbb{Q}(\alpha) : \mathbb{Q}$ where $\alpha$ is the real ...
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Issue in the proof showing that the splitting fields are normal.

I am using the following definition of a normal extension: $K \subset L$ is normal if for all $\Omega$ with $K \subset \Omega$, for all $K$ embeddings $x_1:L\rightarrow \Omega, x_2:L\rightarrow \Omega$...
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Proving that $\mathbb{Q}(\omega)$ is the splitting field for $f(x)=x^4 - x^2 + 1$ over $\mathbb{Q}$

I have been given that $\omega = e^{\pi i\over{6}}$, a 12th root of unity. I have shown that $\omega$ is a root of the polynomial $f(t)=t^4 -t^2 +1$, as are $\omega^5,\omega^7,\omega^{11}$ and that $f$...
Jacob's user avatar
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If F is normal over E and E is normal over K then F need not be normal over K

The following question was given as homework by the prof. and I am not able to deduce it. So, I am asking for help here. If F is normal over E and E is normal over K then prove that F need not be ...
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1 vote
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Intersection of an Indexed Family of normal field extensions is also normal

Let $(L_i)_{i \in I}$ be an indexed Family of normal field extensions over $K$ ($L_i \subseteq \overline{K}$ for an algebraic closure of $K$). Show that $\bigcap_{i\in I} L_i =: L$ is also a normal ...
Shuster's user avatar
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1 answer
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Showing an extension of degree 2 is normal in a different way

Let $F$ be the splitting field of a separable polynomial over $K$, and let $E$ be a subfield between $K$ and $F$. Show that if $[E:K]=2$, then E is the splitting field of some polynomial over $K$. I'...
JayD's user avatar
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$E/L$, $L/K$ normal then $E/K$ normal

I have to decide if the following statement is true: If $E/L$, $L/K$ are normal, then $E/K$ is normal I have though of this proof but I don't know if it is correct, could someone help me? Let $p(x) ...
kubo's user avatar
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Extension of irreducible ideal of a Noetherian ring is irreducible of rings of fractions

Let $S$ be a multiplicatively closed subset of the commutative Noetherian ring $R$ and $f:R \to {S^{ - 1}}R$ denote the natural ring homomorphism. Let $I$ be an irreducible ideal of $R$ for which $S \...
Thắng Cà Mau's user avatar
1 vote
1 answer
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A problem to show that a certain field extension is not normal

Problem: Let $\alpha$ be a real number such that $\alpha^4=5$. Then show that $\mathbb Q(\alpha +i\alpha) $ over $\mathbb Q$ is not a normal extension.(where $i^2=1$) My approach: I could show that ...
Noob mathematician's user avatar
4 votes
1 answer
243 views

Restriction of the Frobenius automorphism for normal extensions

I'm studying number theory on Marcus book and at a certain point I'm required to prove the following facts about the Frobenius automorphism. We start with a lemma and then are required to specialize ...
Alain Ngalani's user avatar
1 vote
0 answers
215 views

Every algebraic extension of a finite field is a finite extension. True or False? [duplicate]

If $F$ is an algebraic extension of a finite field $K$, then $F/K$ is separable. If we are able to show that $F/K$ is normal, then $F/K$ would be a Galois extension and hence splitting field of a ...
Promit Mukherjee's user avatar