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Questions tagged [normal-extension]

For questions about normal field extensions.

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Normal Closure of an Algebraic Extension

$\bf{Q.}$ Let $K/F$ be an algebraic extension. Show that there is an algebraic extension $L/K$ such that $L/F$ is normal and if $M$ is another normal extension of $F$ such that $F\subseteq K\subseteq ...
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Prove if two normal extensions over a field are isomorphic

Suppose $K_1,K_2$ are normal extensions over a field $F$ . Suppose the set of minimal polynomials of elements in $K_1$ over $F$ = the set of minimal polynomials of elements in $K_2$ over $F$ . Then ...
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Help understanding how to show a field extension is a Galois extension .

I have the field extension $\Bbb Q(\sqrt[8]{2},i)$ over $\Bbb Q$. I want to show that this is a Galois extension. I know that I can do this by showing that It is an extension which is both normal and ...
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Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal. If we let $x = \sqrt[4]{11}+i\sqrt[4]{11} = \sqrt[4]{...
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What automorphism cannot be extended when the extension is not normal?

As I understand it, if we have an automorphism $\phi : K \rightarrow K$, and a finite normal extension $N/K$, $\phi$ can always be extended to an automorphism of $N$. But what happens if $N/K$ is not ...
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Which extensions of $Q$ with $\sqrt{a+\sqrt{5}}$ are normal? [duplicate]

Find all rational numbers $a$ such that $\mathbb{Q}(\sqrt{a+\sqrt{5}})$ is a normal extension of $\mathbb{Q}$. Via the polynomial $(x^2 - a - \sqrt{5})(x^2 - a + \sqrt{5})$ we can see that the ...
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Finding whether an extension is normal

I have $\mathbb Q(i,\sqrt5)$ and i think i found the base for the extension on $\mathbb Q$ as ${ a+b\sqrt5 +ci + di\sqrt5 ; a,b,c,d in \mathbb Q}$ But i don't know at which polynom to look in order to ...
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Are normal extensions always simple?

I'm trying to diagram out the relations between simple, normal, separable, and Galois field extensions in my head. I understand that all separable extensions are simple, and Galois extensions are ...
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1answer
30 views

Intermediate Fields and Root Fields

Suppose that $F \subset K \subset E$ are fields and $E$ is the root field of some polynomial in $F[x]$. Show, by means of an example, that $K$ need not be the root field of some polynomial in $F[x]$. ...
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$E/k$ is normal iff $E^{Aut(E/k)}/k $ is purely inseparable?

I came to think of it after reading the proposition about decomposition of normal extension into tower of purely inseparable extension and separable extension in Lang's Algebra. I tried to prove that ...
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smallest normal extension containing an infinite algebraic extension

Let $F/k$ be an algebraic extension. Let $S(F/k)$ be the set of all embeddings of F over k into algebraic closure $k^\mathrm{a} $. I'm trying to prove that the smallest normal extension of k ...
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non separable, non normal, finite field extension

I want to give an example of a non-separable, non-normal, finite field extension. So I have to find an extension of a field, which is not perfect. I would suggest $K=\mathbb{F}_2(t)$. An extension, ...
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Property of Splitting Fields and K-homomorphisms

Let $K \subset \Bbb C$ be a field, let $P \in K[X]$ be a polynomial of degree $n$ and let $N$ be the splitting field of $P$ over $K$. Then $N$ is a normal extension of finite degree over $K$. The ...
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1answer
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Normal extensions equivalence, Morandi Galois Theory, p36, Theorem 3.28

If $K$ is algebraic over $F$, then the following are equivalent: (2) If $M$ is an algebraic closure of $K$ and if $\tau: K \rightarrow M$ is an $F$-homomorphism, then $\tau(K)=K$. (3) If $...
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Can I prove that a splitting field is normal without using zorn lemma

There is a theorem : If $K \in F$ and $F$ is a splitting field of a polynomial in $K[x]$,then F is a normal extension over $K$. For proving this I choose a polynomial $g \in K[x]$ which has a root ...
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420 views

The relationship between normal extension and the splitting field of polynomial

An algebraic extension $K/F$ is called a normal extension if any irreducible polynomial of $F[x]$, which has a root in $K$, can be completely decomposed in $K$. I have know that a finite extension $K/...
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63 views

Example of an extension where Normal of a Normal extension is Normal.

We know that in general Normal extension of a Normal extension may not be normal. I want an example where it holds, i.e Normal extension of a Normal extension is still Normal. Is there any example of ...
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1answer
42 views

If $E/K$ is normal and $K \subseteq F \subseteq E$ then $E/F$ is normal.

Assume that $E/K$ is a normal extension. Given the tower $K \subseteq F \subseteq E$ proof that $E/F$ needs to be normal. I would like to use the characterization of normal extensions $F/K$ that ...
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1answer
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“Intersection” of separable subfields [duplicate]

I have the following question, from Isaacs' Algebra book. Suppose $F\le E$ is a finite-degree normal field extension, and that $K$ and $L$ are intermediate subfields (between $F$ and $E$). If $E$ is ...
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Is there a polynomial over $\mathbb{Q}$ which $K$ is its splitting field.

Let $K=\mathbb{Q}(\sqrt[11]{7},i)$. Is $K$ the splitting field of some polynomial over $\mathbb{Q}$? My attempt: My first intuition would be no. Since if $K$ is the splitting field of some ...
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Kaplansky's vs common definition of normal field extension

Kaplanksy's definition of normal field extension goes as follows Let $K\subseteq E$ be fields. $E$ is normal over $K$ if for any $\alpha\in E\setminus K$ there exists $\psi\in \operatorname{Aut}_K(...
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Questions regarding tower of normal/separable extensions

I am learning about Galois theory these days. And I am considering to prove: Is that the fact that given a tower of extensions $A/B/C/D$, if $A/B$, $B/C$, $C/D$ are normal, then $A/D$ is normal? Is ...
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How to determine whether an extension is normal extension.

Let $\alpha = e^{2 \pi \imath/19} + e^{16 \pi \imath/19} + e^{14 \pi \imath/19} + e^{36 \pi \imath/19} + e^{22 \pi \imath/19} + e^{24 \pi \imath/19}$, its minimal polynomial is $f(x) = x^3 + x^2 - 6x -...
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3answers
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Totally transcendental extension and irreducibility.

I want to prove the theorem below: Theorem. Let $F \subseteq E$ be a totally transcendental extension and let $f\in F[X]$ be irreducible. Show that $f$ is irreducible in $E[X]$ I could not go ...
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1answer
106 views

$c$ be algebraic , constructible over $\mathbb Q$ , $N$ be the normal closure of $\mathbb Q(c)/\mathbb Q$ ; then $[N:Q]$ is a power of $2$?

$c$ be algebraic over $\mathbb Q$ , $N$ be the normal closure of $\mathbb Q(c)/\mathbb Q$ ; if $c$ is constructible over rationals then how to show that $[N:\mathbb Q]$ is a power of $2$ ?
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$L/k$ finite extension , $L_1,L_2 $ subfields of $L$ containing $k$ , $L_1/k$ separable and $L_2/k$ normal , then $[L_1L_2:L_2]=[L_1:L_1\cap L_2]$ ?

Let $L/k$ be a finite extension . $L_1,L_2 $ subfields of $L$ containing $k$ such that $L_1/k$ is separable and $L_2/k$ is normal . Then it is easy to see $L_1L_2/L_2$ is separable . But how to show ...
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For which rational values of $a$ is $\mathbb{Q}(\sqrt{\sqrt{5} + a}) / \mathbb{Q}$ a normal extension?

For which rational values of $a$ is $\mathbb{Q}(\sqrt{\sqrt{5} + a}) / \mathbb{Q}$ a normal field extension? My main approach to solving this has been to look at when $\mathbb{Q}(\sqrt{\sqrt{5} + a})$...
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A statement about normal basis element , trace and characteristic of field

Let $F/K$ be a normal finite extension where $F=F_1 \times F_2$ for subfields $F_1,F_2$ of $F$ where $F_1\ne F$ . Suppose $w$ is a normal basis element in $F/K$ for any $w\in F$ for which $Tr_{F/F_2}(...
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A finite extension of a normal extension is normal?

I'm dealing with the question. Let char$K=0$ and $F/K$ be a finite and normal extension. Now, given $g(x)\in K[x]$ and $L$ be the splitting field of $g(x)$ over $F$. Show that $L/K$ is a normal ...
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850 views

In a normal extension of a field, is there an automorphism that maps irreducible factors of a certain irreducible polynomial?

Let $F$ be a field, $f(x)$ be an irreducible polynomial in $F[x]$ and $E/F$ be a normal extension. Show that if $g(x)$, $h(x)$ are irreducible factors of $f(x)$ in $E[x]$ then there exists an ...
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Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
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1answer
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Normal extensions problem in Lang

This is a problem in Lang's Algebra. $F$ is finite normal extension over $k$ and $f(x)$ is irreducible in $k[x]$. If $f(x)=g(x)h(x)k(x) \in F[x]$ where $g(x),h(x)$ are monic irreducible factors in $F[...