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Questions tagged [galois-extensions]

For questions about Galois extensions fields. We say $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

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Galois Extensions

I was trying to answer the following question and wasn't sure how to proceed. Which one of the following extensions $K \subset L$ is not Galois? (a) $K = \mathbb{Z}_3(x)$ and $L = K[a]/(a^3-x)$ ...
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A problem about the field of rational functions over finite field

Let $p$ be a prime number, and let $F_{p}$ be the finite field with $p$ elements. Let $F=F_{p}(t)$ be the field of rational functions over $F_{p}$ . Consider all subfields of $F$ such that $F/C$ is a ...
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Condition on Galois Group that makes polynomial irreducible

Suppose $K$ is the splitting field of a monic polynomial $f(x) \in \mathbb{Z}[x]$. What is the condition on $G = \text{Gal}(K/\mathbb{Q})$ that ensures $f$ is irreducible?
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A question regarding the Galois extension of the cyclotomic polynomial $\Phi_{15}$

Given the cyclotomic polynomial $\Phi_{15}$, I am trying to : i) Determine the isomorphism type of the Galois group of $\Phi_{15}$ over $\Bbb Q$. ii)Letting ω be a primitive 15-th root of unity in $\...
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Show that $E/F$ is Galois extension

If $F$ has characteristic $\neq$2 and $E/F$ is a field extension with $[E:F]=2$, then $E/F$ is Galois. Normal and separable extension is Galois extension. Can we say that since the degree of ...
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Find all subfields of $\mathbb{Q}(\mu_{24})$

Problem: Let $\mu_{24} \in\mathbb{C}$ be a primitive 24'th root of unity and let $L = \mathbb{Q}(\mu_{24})$ be the 24'th cyclotomic extension of $\mathbb{Q}$. List all subfields of $L$ in the form $\...
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Group is soluble if and only if quotient is abelian

So a group G is soluble if and only if it has a subnormal series $$ \{ 1\} =G_0 \ \triangleleft \ G_1 \ \triangleleft \ ... \ \triangleleft \ G_n=G $$ where all quotient groups $G_{i+1}/G_i $ are ...
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My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

I'm trying to understand Galois theory and any help on this question I'm working on would be very much appreciated. Let $E$ be the splitting field of $x^3-5$ over $\Bbb Q$. Compute $\mathrm{Gal}(E:\...
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Efficiently calculating the Galois group of $p(x)=x^4+4x^2-2$

I need to find the Galois group of $p(x)=x^4+4x^2-2$. Here is what I have done so far: By Eisenstein's criterion, $p$ is irreducible over $\Bbb Q$. Therefore, the Galois group is a transitive ...
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If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

A lemma in my lecture notes states that if $K$ is a subfield of $\mathbb{C}$ and $\zeta=\exp(2\pi i/p)$ then $K(\zeta)/K$ is Galois. They proved it by arguing that the minimal polynomial of $\zeta$ ...
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Suppose $a,b\in E$ and $\sigma\in \textrm{Gal}(E/F)$ with $\sigma(a)=b$. Show that $a$ and $b$ have the same minimal polynomial in $F[x]$

Let $E/F$ be a Galois extension. Suppose $a,b\in E$ and $\sigma\in \textrm{Gal}(E/F)$ with $\sigma(a)=b$. I need to show that $a$ and $b$ have the same minimal polynomial in $F[x]$. Here's what I ...
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The Galois group of $x^4+3x^3-3x-2$ over $\mathbb{Q}$.

I am trying to find the Galois group of $x^4+3x^3-3x-2$ over $\mathbb{Q}$ which is a problem from Dummits and Foote(Exercise 14.6.6). and got stuck. What I have tried is as following. Let $p(x)=x^...
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Show that $f(x)$ is a power of $\operatorname{irr}(α, F)$, and $f(x) = \operatorname{irr}(α, F) \iff K = F(α)$.

Let $K$ be a finite normal extension of $F$. $f(x)=\prod_{\sigma \in G(E/F)} (x − \sigma(\alpha))$ where $f(x)\in F[x]$ How can I show that $f(x)$ is a power of $\operatorname{irr}(α, F)$ and $f(x) =...
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Let $K/E$ and $E/F$ be Galois extensions. If every $σ ∈ \text{Aut}(E/F)$ is the restriction of an element of $\text{Aut}(K/F)$, then $K/F$ is Galois.

$\newcommand{\Q}{\mathbb{Q}}$$\DeclareMathOperator{\Aut}{Aut}$Let $K/E$ and $E /F$ be Galois extensions. I would like to show that, if every $\sigma \in \Aut(E/F)$ is the restriction of an element of $...
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The degree of extension

I have two extensions and must find their degrees: a) $\mathbb{C}:\mathbb{Q}$; b) $\mathbb{R\{5}\}:\mathbb{R}$. I know that a) degree is infinity and b) is 1. It for me seems trivial, but how it ...
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Automorphism of $\mathbb{C}$ over a subfield $K$ of $\mathbb{C}$ [duplicate]

Assume a finite field extension $\mathbb{C}/K$ such that $[\mathbb{C}:K]>2$. Let $\varphi \in \text{Aut}(\mathbb{C}/K)$, so $\varphi \in \text{Aut}(\mathbb{C})$ and $\varphi\vert_K=\text{id}\vert_K$...
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Primitive Element for Splitting Field of $x^3-2$?

Let $\alpha:=\sqrt[3]{2}\in\mathbb{R}$ and $\omega:=e^{2\pi i/3}\in\mathbb{C}$. Then the splitting field for the polynomial $x^3-2\in\mathbb{Q}[x]$ is $$\mathbb{Q}(\alpha,\omega\alpha,\omega^2\alpha)=\...
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Why a homomorphism from full Galois group of rationals cannot factor.

I need to fully understand why if I have epimorphism $f: G_{\Bbb Q}\to \Bbb Z/2\Bbb Z\;$ , with $\;\Bbb G_{\Bbb Q}=\;$ full Galois group of rationals $\;\Bbb Q\;$ , and when last group corresponds ...
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1answer
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Galois Group and Subfield lattice of $(x^2 - 7)(x^4 + x^3 + x^2 + x + 1)$ over $\mathbb{Q}$

If $K$ is the splitting field of $f(x)$, then $K$ is Galois Closure of $Q(\sqrt{7}, \zeta_5)$ where $\zeta_5 = \alpha$ is a primitive root of unity. There's 6 roots of $f(x)$: $\pm \sqrt{7}, \alpha, \...
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Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?

Exercise sounds: Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?I have the solution (on picture). Is it correct? Why do we prove in this way, why we must show that square, ...
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Prove that the Galois group of $f\left(x\right)$ over $\mathbb{Q}$ is not a simple group.

Let $f\left(x\right)\in\mathbb{Q}\left[x\right]$ be an irreducible polynomial of degree $n>2$ which has $n-2$ real roots and exactly one pair of complex roots. Prove that the Galois group of $f\...
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Show that the minimal polynomial over $F$ of each element of $E$ has only one distinct root.

Let $K$ be a finite normal extension of $F$ and let $E$ be the fixed field of the group of all $F-$automorphisms of $K$. Show that the minimal polynomial over $F$ of each element of $E$ has only one ...
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Let $M$ be a subfield of Complex field such that $M/\Bbb Q$ is a finite Galois extension

Let $M$ be a subfield of Complex field such that $M/\Bbb Q$ is a finite Galois extension. Show that if $[M:\Bbb Q]$ is an odd number, then $M$ is a subfield of Real field. My current thought is since ...
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Method of showing maximal number of intermediate field extensions of a Galois extension with given degree

The task is the following: Show that a Galois extension $L/K$ of degree $45$ has got at most $12$ intermediate field extensions. Below I present a proof. I seek a more general method for this ...
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Extension of degree 5 not obtained by adjoining a 5th root and whose normal closure contains a primitive 5th root of unity.

The question is mainly what the title says, but here is the setup in more details. Let $K$ be a field not containing a primitive 5th root of unity (for this question, the case $K = \mathbb{Q}$ seems ...
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How to calculate the Galois group of $x^5+15x+12$?

How to calculate the Galois group of $x^5+15x+12$ over the field $\Bbb Q$? Using the Tchebotarov Density theorem which states that "the density of primes $p$ for which $f(x)$ splits into type $T$ ...
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$Gal(f)$ is cyclic of order $m$ for irreducible $f \in K[X]$ of degree $m$, where $K$ is a finite field

Let $K$ be a finite field, $f \in K[X]$ irreducible with degree $m$. Show that $Gal(f)$ is cyclic of order $m$. I have shown that $f$ is separable over $K$ by using that $K$ is finite and thus ...
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Meaning of a symbol regarding field.

I was going through a problem in Dummit and Foote's Abstract Algebra. That problem involves two subfields $K$ and $E$, and $K=Q(a^{1/n})$. In that problem they have a hint that $N_{K/E}(a^{1/n})\in E$....
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Splitting field of separable polynomial is Galois extension

Definitions: $f$ is separable if every irreducible factor has distinct roots. $E/F$ is a Galois extension if the fixed field of the Galois group Gal$(E/F)$ is $F$ I would like to prove the following ...
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Question regarding Galois group and algebraic extensions

From Dummit and Foote I was reading about cyclotomic extensions, where I came across the definition of algebraic extension saying the extension $K/F$ is an algebraic extension if $K/F$ is Galois and $...
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Does $\mathbb{F}_9$ contain a 4th root of unity?

I realised that I don't know how to construct $\mathbb{F}_9$. I'm guessing that $\mathbb{F}_9 = \mathbb{F_3(\theta)}$, where $\theta$ is the root of some irreducible polynomial over $\mathbb{F}_3[x]$ ...
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Example of a Galois extension $L/\mathbb{Q}$ with $\text{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_{4}$

I propose that $L=\mathbb{Q}(i\sqrt[4]{2})$. Obviously $\mathbb{Q}(i\sqrt[4]{2})$ is the splitting field of $f=t^{4}-2\in\mathbb{Q}[t]$, since $N:=\{\text{zeros of $f$}\}=\{\sqrt[4]{2},-\sqrt[4]{2},...
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Show that $f(x)$ factors in $K[x]$(whether or not $K$ is contained in the Galois closure $L$ of $f(x)$).

Let $f(x)\in F[x]$ be an irreducible polynomial of degree $n$ over the field $F$. For any $K$ is any Galois extension of $F$, show that $f(x)$ factors in $K[x]$ see this (whether or not $K$ is ...
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Proving that $[\mathbb{Q}(\sqrt{\sqrt{p+q}+\sqrt{q}},\sqrt{\sqrt{p+q}-\sqrt{q}}):\mathbb{Q}]=8$.

Some days ago I posted a question in MSE in order to correct a solution to the problem of Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}]=8$. After posting this another ...
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Is it true that $Gal(K/F)\cong S_{n_1}\times \cdots S_{n_k}$?

I was reading galois theory and galois group from Dummit Foote and while reading Galois groups of polynomial a sudden question came into my mind that if $f(x)$ is an irreducible separable polynomial ...
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Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
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Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
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Normal Subgroup of Galois Group

Let $L/K$ be a Galois extension, and let $R\subseteq L$ be a subring such that $\tau(R)=R$ for every $\tau\in\text{Gal}(L/K)$. Let $\alpha\in R$. How would I show that $H=\{\tau\in\text{Gal}(L/K):\...
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cyclic extension of prime power of a local field

Let $K$ be a non archimedian local field of characteristic $p>0$ with residue field $\mathbb{F}_p$ and $l\neq p$ be a prime. It is known by local classfieldtheory that any abelian Galois ...
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Degree of extension - Is this $\phi(n)$ or $\phi(n)/2$?

Let $a=\cos(2\pi/n) $. I have shown that $Q(a)/Q$ is a Galois extension and now I want to show that $[Q(a):Q]=\phi (n)/2$. I have done the following: It holds that $|Gal(\mathbb{Q}(a)/\mathbb{Q})...
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Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
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An extension $K/F$ is Galois if for every simple extension $F \subset F(u) \subset K$, $[F(u):F] \leq 2$.

Let $F$ be a field, $char(F)\neq 2$ and let $K$ be an extension field of $F$. If for each $u\in K$, $[F(u):F]\leq 2$, show that $K$ is a Galois extension of $F$.
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Determine a Normal Basis for Galois Extension of $\mathbb{Q}$ with primitive pth root of unit (p prime)

Let p be a prime, $\xi_p \in \mathbb{C}$ a primitive p-th unit root and $K = \mathbb{Q}(\xi_p)$. Give a normal basis for $K/\mathbb{Q}$. I know, that a basis of $L/K$ (finite and galois) is ...
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Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$?

In the proposition "The polynomial $x^{p^n} -x$ is precisely the product of all the distinct irreducible polynomials in $\Bbb F_p[x]$ of degree $d$ where $d$ runs through all divisors of $n$." Can we ...
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1answer
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Any Sub-extension of a Radical Extension is Solvable?

I have seen the following theorem stated without proof. (I assume that all fields are characteristic zero.) Theorem: Let $F\subset K$ be a radical extension of fields. That is, suppose that $K$ can ...
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Is this a counterexample?

Suppose $K $ is a field and $\overline K $ an algebraic closure. Let $f $ be a $K $-automorphism of $\overline K$, let $L$ be the subfield of $\overline K $ fixed by $f $. In this post : (link), they ...
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1answer
52 views

Galois group is determined by action on roots of polynomial [duplicate]

This is a very simple question but I can't give a good answer to it. If we have a field $K$ and a Galois extension $L/K$ where $L=K(\alpha_1,\ldots,\alpha_n)$ and $\alpha_1,\ldots,\alpha_n$ are the ...
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0answers
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Question regarding Galois theory

While reading a theorem on Galois theory I came up with this statement "Let $E/F$ be a finite separable extension. Then $E$ is contained in an extension of $K$ which is Galois over $F$. Now my ...
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0answers
49 views

Primes lying above a prime in a Galois extension

Assuming $L$ is Galois over $K$, and that $O_L$ and $O_K$ are their respective rings of integers. Let $p$ be a prime ideal of $O_K$, is there a classification of the prime ideals laying above $p$ in $...
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1answer
59 views

Compute the degree of the splitting field of $x^{12} - 2$ over $\mathbb{Q}$ and describe its Galois Group as a semidirect product

I have the polynomial $f(x) = x^{12} - 2$. I have to compute the degree of the splitting field over $\mathbb{Q}$ and describe its Galois Group as a semidirect product. Clearly the splitting field is $...