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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Abstract norm map in Neukirch's book

At page 277 of Neukirch's ANT is defined the abstract norm map and the norm residue group, but I have a problem that I'll explain below: $G$ is a profinite group and the closed subgroups of $G$ are ...
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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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$K(x)$ 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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75 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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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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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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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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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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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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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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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
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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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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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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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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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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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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
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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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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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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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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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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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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
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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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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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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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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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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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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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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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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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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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 ...
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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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Monic irreducible polynomials of degree 6 in $F_{5}[X]$

Question A How many monic irreducible polynomials of degree 6 in $F_{5}[X]$ Question B Give an example of an irreducible polynomial of degree 6 in $F_{5}[X]$ Idea for a Such a polynomial would be ...
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General technique for finding minimal polynomial? [closed]

I always have a lot of trouble with these problems, "find the minimal polynomial of {number} over {field}" What are the general procedures for solving problems of this format? Thank you for your ...
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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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Galois Group of a complex polynomial

How I can find $|Gal(f(x))|$ with $f(x)=x^2+2ix+1$? I know that roots are $x_1=-i+i\sqrt{2},x_2=-i-i\sqrt{2}$ If $L:=\mathbb{Q}(x_1,x_2)$, then ...
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roots of multi-variable polynomials and extension fields

I am teaching a course in (standard single-variable) Galois theory and the following, presumably naive, question occurred to me: Given a finite collection of polynomial equations in a finite number ...
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Minimal polynomial over $\mathbb Q(\sqrt{-2})$

Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$. I have done the first part of this, over $ Q$, and have a polynomial. But I do not know ...
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What is the general theory of solving polynomial equations “beyond radicals”?

For example, using Bring radicals or elliptic functions to solve quintic equations. Wikipedia says that similar methods can be used for higher degree polynomials, but I'm struggling on finding ...
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Find the minimal polynomial over a field

I have two similar questions: 1). Find the minimal polynomial for $a^{-1}$ (a to the power of minus 1) over $F_3$. $a$ is the root of the polynomial $x^3-x+1$ in $F_3[X]$ I have used the division ...
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Why are Galois extensions mapped to themselves?

Let $L$ be the splitting field of a separable irreducible polynomial $f(x)\in\Bbb{Q}[x]$. Consider the Galois subfield $K\subset L$. Let $\sigma\in \text{Gal}(L/\Bbb{Q})$. As $K$ is a galois ...
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38 views

Is every subgroup of automorphisms a Galois group?

Let $G=\{\sigma_1=1, \sigma_2, \dots,\sigma_n\}$ be a subgroup of automorphisms of a field $K$ and let $F$ be the fixed field. Then $[K:F]=|G|$ Why is this always true? I thought this would be ...
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Show that $|\text{Hom}_k(K,\widetilde{K})(\phi)|\le [K:L]$

Let $K/k$ be a finite field extension, $L$ an intermediate field and $\widetilde{K}$ such that $\widetilde{K}/k$ is normal. Let $\phi \in \text{Hom}_k(L,\widetilde{K}) := \{\psi:L\rightarrow ...
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The proof of a major theorem in Galois theory

Let $f(x)\in K[x]$ be a separable polynomial of degree $n$. If $K$ does not have characteristic $2$, the Galois group of $f(x)$ over $K$ is a subgroup of $A_n$ if and only if disc $f$ is a square ...
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Counter example of equivalence between separable polynomial and square-free polynomial

Over a perfect field $k$, the notions of separable and square-free polynomial are equivalent. Indeed if $b^2|P$, then $P$ has a repeated root in $\bar k$. Conversely, if $p$ has a repeated root in ...