Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Basic question in Galois theory (on applying elements of the Galois group to a root of polynomial)

Suppose I have $K = \mathbb{Q}(\theta)$ and let $f$ be the minimal polynomial of $f$ over $\mathbb{Q}$. Suppose $f$ has degree $n$ so that the degree of $K$ over $\mathbb{Q}$ is $n$. Suppose further ...
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1answer
70 views

Finding Galois Group

Let $\mathbb Q\subset\mathbb Q(\sqrt{3}+\sqrt[3]{5})$. I want to find the Galois group of the given field extension. It would be easy for me if I could find a basis of the given field extension ...
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36 views

how to find coefficient c1, c2, c3, c4 of a polynomials of degree 4 from resolvent

if not starting from standard resolvent of each degree and use (y-x1...)(y-x2...)(y-x3...) and group theory how to find corresponding c1, c2, c3, c4 of polynomial x^4+c4*x^3+c3*x^2+c2*x+c1 which c1, ...
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1answer
17 views

$K(x)$ not stable relative to $K(x,y)$ and $K$

Prove that in the extension of an infinite field $K$ by $K(x,y)$, the intermediate field $K(x)$ is Galois over K, but not stable (relative to $K(x, y)$ and $K$). I know that if K(x) is algebraic it ...
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Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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35 views

Splitting field of a polynomial has a solvable group of automorphisms

Prove that splitting field of a polynomial $f \in \mathbb{k}[x]$ ($f = 0$ solvable by radicals) has a solvable group of automorphisms $Aut_\mathbb{k}(\mathbb{F})$. I have just started learning Galois ...
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1answer
42 views

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$ over the field $\mathbb{k}_{sym}(x_1,\dots,x_n)$, the characteristic of $\mathbb{k}$ is 2 and $$D(x_1,\dots,x_n) = \prod_{1 \leq ...
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3answers
46 views

Local reciprocity map applied to norm

This question concerns the local reciprocity homomorphism $r_L : L^\times \to G_L^\text{ab}$, where $L$ is a local field with absolute Galois group $G_L$. If $K\subset L$ is a subfield and $r_K$ is ...
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1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
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1answer
47 views

Galois group of $\mathbb Q(\sqrt{4+\sqrt 7})/\mathbb Q$

Let $\alpha =\sqrt{4+\sqrt 7}$ and $\beta =\sqrt{4-\sqrt 7 }$. I have to compute $Gal(\mathbb Q(\alpha )/\mathbb Q)$. So, I found that $Gal(\mathbb Q(\alpha )/\mathbb Q)=\mathbb Z/2\mathbb Z\times ...
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1answer
28 views

Prove that $f = q_1 + Gq_2$ for some $q_1, q_2 \in \mathbb{k}_{sym}(x_1,\dots,x_n)$

Let the orbit of the function $f \in \mathbb{k_(x_1,\dots,x_n)}$ under the action of $\{\phi_\sigma|\sigma \in \mathfrak{S}_n\}$ has length 2. Prove that $f = q_1 + Gq_2$ for some $q_1, q_2 \in ...
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1answer
95 views

Is $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ a subfield of $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ a subfield of $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I think that $\sqrt{2}+\sqrt{3}$ since $\sqrt{2}, \sqrt{3} \in \mathbb{Q}(\sqrt{2},\sqrt{3})$. So maybe ...
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1answer
40 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
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1answer
32 views

Describe the group of $\operatorname{Aut}_{K(x^p, y^q)}(K(x, y))$

Let $K$ be a field, $\operatorname{char}K = p > 0$, $q$ is prime such that $p-1 \equiv 0 \pmod{q}$. Describe the group of $\operatorname{Aut}_{K(x^p, y^q)}(K(x, y))$. It's pretty clear that we ...
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Find roots or splitting field of a polynomial given its Galois group

A well known result from Galois theory is that the roots of a polynomial can be expressed by a formula using field operations and taking $k$-th roots if and only if the Galois group of the polynomial ...
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1answer
40 views

Suppose $\phi\in{Aut(C)}$ and continuous, why $\phi$ must fix $R$? [closed]

Suppose $\phi\in{Aut(\mathbb{C})}$ and continuous, why $\phi$ must fix $\mathbb{R}$? I know that continuity is crucially important here, but I'm not sure how.
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1answer
55 views

Does Hilbert 90 need the extension to be Galois?

I read about Hilbert 90 in Morandi's Fields and Galois Theory (GTM 167). Let $K/F$ be a finite Galois extension. Let $G=\operatorname{Gal}(K/F)$. Define a crossed homomorphism $f\colon G\to K^*$ ...
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1answer
68 views

Galois group for Kummer extension over Cyclotomic extension of $p$-adic field

I am trying to recover the Galois group of the extension $E/F$, where $E$ and $F$ are the fields defined below. $F$ is a finite extension of $\mathbb{Q}_p$, containing a primitive root of unity ...
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74 views

composition of a $\mathbb{Z}_p$-extensions and Hilbert class field extension (Iwasawa)

I am reading Iwasawa's article On $\Gamma$-extensions of algebraic number fields. In paragraph 7.3 : " We now take as M the maximal unramified abelian $p$-extension of $L$ in $\Omega$ , i.e. the ...
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Find isomorphism between Kummer's field and filed of n-th roots

Let $\mathbb{k}$ be the field of all the $n$th roots of $1$ and $\mathbb{F=k}(\alpha_1, ..., \alpha_n)$ is a Kummer's field over $\mathbb{k}$, where $\alpha^n_i=a_i \in \mathbb{k^*}$, $i=1,\dots,m$. ...
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Finite separable normal extension has Galois abelian group

Prove that finite separable normal extension $\mathbb{F}$ of field $\mathbb{k}$, $\operatorname{char}(\mathbb{k})=p > 0$ has Galois abelian group $\operatorname{Gal}(\mathbb{F}/\mathbb{k})$ with ...
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1answer
22 views

Can I find a Galois extension which contains a finite set of algebraic elements?

Suppose $K/F$ is an algebraic extension of fields. Pick a finite collection $a_1,...,a_n \in K$. Can I find a Galois extension of $F$ which contains these $n$ elements of $K$?
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1answer
32 views

Outer Automorphisms of Galois groups

Assume $K/\mathbb{Q}$ is Galois extension of number fields and let $F \subset K$. Let $Gal(K/F) \cong G$ and let $\sigma$ be an outer automorphism of $G$ as an abstract group. Is there a way to ...
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1answer
52 views

Is the tensor product of 2 finite extension of $\Bbb Q$ isomorphic to a direct sum of fields?

I have $K_1$ and $K_2$ two finite extensions of $\Bbb Q$. I can construct $K_1 \otimes_\Bbb Q K_2$. This is clearly isomorphic to a direct sum of field as vector space (indeed one can easily see that ...
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2answers
61 views

Isn't the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}(\sqrt3))$ just the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2))$?

Isn't the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}(\sqrt3))$ just the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2))$? I'm getting confused about common algebra notation. ...
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1answer
73 views

Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong ...
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48 views

Minimal Galois extension, describe structure of $Gal(L/\mathbb Q)$

Find the minimal Galois extension $L$ of $\mathbb Q$ containing $\mathbb Q(\sqrt[4]{5})$. Describe the structure of $Gal(L/\mathbb Q)$. I think $L$ is a splitting field of $X^4-5$ over $\mathbb Q$. ...
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$\mathbb{Z}[ \sqrt{−n}]$ is not a UFD [duplicate]

a) $\mathbb{Z}[ \sqrt{−n}]$ is not a UFD. I proved for $n=3$, $n=5$ that in $\mathbb{Z}[\sqrt{−n}]$ we have that 2 is irreducible but not prime. But how can I prove that in general, for $n \geq ...
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how many orbits are possible in the group action?

Let G be Galois group of a field with nine elements over its Subfield with three elements.Then the number of orbits for action of G on the field with nine elements?
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1answer
38 views

Multiplicative Inverse in a $256$ Galois Field

I am working on finding the multiplicative reverse in $GF(2^8)$ using the Euclidean Algorithm but after reading multiple sources, I feel as though I am proceeding incorrectly. Using the irreducible ...
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2answers
49 views

Why are the elements of a galois/finite field represented as polynomials?

I'm new to finite fields - I have been watching various lectures and reading about them, but I'm missing a step. I can understand what a group, ring field and prime field is, no problem. But when we ...
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1answer
77 views

Abelian extensions with squarefree discriminant

Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many? Some even degrees may be ...
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1answer
411 views

degree of the extension galois

I have a problem with the solution of the tasks of abstract algebra.Help please. $F=({\bf Q};+;\cdot),K=({\bf R},+;\cdot)$. Determine the degree of the extension $F_K^* (\sqrt 2,\sqrt 3):F]$. ...
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20 views

Trisecting angle equivalence of constructing a segment

After reading Wikipedia and some previous questions asked in this site, I still don't understand this. Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). ...
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2answers
28 views

Finite extensions of $\mathbb F_p(t)$

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$. Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some ...
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32 views

Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where ...
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1answer
25 views

Completion of a Galois Extension $K | \mathbb{Q}$ w.r.t. to a non archimedean abs. value is a Galois extension

As the title suggests, I'm interested in a proof of the following fact: Let $K | \mathbb{Q}$ be a finite Galois extension, and let $\mathfrak{p}$ be a prime ideal of the ring of integers of $K$. ...
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2answers
46 views

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...
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How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt ...
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1answer
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Basic questions about field homomorphism extension

I learned that one can extend the homomorphism "injection" $k\hookrightarrow \Omega$ (algebraic closure) to a morphism $u:k[a]\to \Omega$ where $a\in \Omega$ is algebraic over $k$ such that the ...
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Minimal cyclotomic field containing a given quadratic field?

There was an exercise labeled difficile (English: difficult) in the material without solution: Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural ...
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$V\otimes_kK\cong \oplus We_i$?

Let $K/k$ be a Galois extension with Galois group $G$, let $V$ be a $K$-vector space with semi-linear $G$ action, which means: $G\to Aut(V_k)$, $g\mapsto r_g$ such that $r_1=Id, r_{gh}=r_g\circ r_h$, ...
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1answer
36 views

Finding Galois conjugates

I'm working on a big exercise from Dummit & Foote (p.584) with the end goal of constructing a polynomial with Galois group $Q_8$ (Quaternion group of order $8$). Take $$\alpha = ...
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2answers
63 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
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101 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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1answer
108 views

Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$

For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$? This problem appears to be quite common, however my understanding of Galois ...
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2answers
38 views

Show that sum of roots is rational.

If $f(X)$ in $\mathbb{Q}[X]$ is an irreducible polynomial polynomial of degree $n \geq 2,$ with roots $\alpha_1, \alpha_2,\ldots,\alpha_n$ in $\mathbb{C},$ show that $\displaystyle ...
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1answer
45 views

Intersection of Kummer extension [closed]

Let $p$ and $q$ be two prime numbers and $\omega$ be the primitive 3rd root of unity. The splitting field of $X^3-p$ over $\mathbb{Q}$ is $K_p=\mathbb{Q}(p^{\frac{1}{3}},\omega)$ and we have a similar ...
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1answer
31 views

Galois group of a polynomial over $\mathbb{C}[t]$

To find the Galois group of the polynomial $X^3-X-t\in\mathbb{C}[t]$, an approach is to compute the discriminant (equal $(2-\sqrt{27}t)(2+\sqrt{27}t)$) which is not a square in $\mathbb{C}[t]$ so the ...
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1answer
28 views

Invariant subfields and Galois group

Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Let $\alpha\in K$ be a root of $f$. Let $H$ be the subgroup of $G$ that ...