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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Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
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Field Extensions of Q by radicals

Is Q(√6) = Q(√3,√2)? I understand that the degree of these field extensions comes from the degree of the minimal polynomial and (alternatively) basis of the field extension. I know that the basis of ...
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The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Hungerford -Algebra p.271 Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$. Let ...
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Galois extension of degree $ 2^n $

I'm trying to find a way to prove the following statement: Assume $ \mathbb{Q} \subset E $ is a Galois extension of degree $ 2^n $. Show that there are fields $ \mathbb{Q} = E_0 \subset E_1 \subset ...
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Galois group of $ x^n - a $ over a field containing $ \zeta_n $

I'm having trouble solving an exercise regarding Galois theory. Suppose $ n > 0 $ and $\zeta_n \in F \subset \mathbb{C} $, where $ \zeta_n $ denotes the primitive root of unity of degree $ n $. ...
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Galois group and intermediate fields for splitting field of $ x^3 -7 $

I'm trying to do the following exercise: find the Galois group $ G(E/\mathbb{Q}) $, where $ E $ is the splitting field of $ x^3 - 7 $, all its subgroups and the intermediate subfields $ E^H $ ...
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Galois extension of degree 75

I'm studying Galois Theory and I stumbled upon this question: Let $E/K$ be a Galois extension of degree $75$. Show that there exists an intermediate subfield $K \subsetneq F \subsetneq E$ such ...
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Kummer extensions, but then a bit more

I'm struggling with the following past exam question, parts (b) and (c): So, starting with (b): since $E/F$ is Galois of degree $p$, we know that $$\Gamma(E/F)\cong C_p\cong ...
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Construction of non-prime finite fields

I am new to Galois field theory and I am struggling with some definitions. To construct any non-prime finite field $GF(p^n)$ with p prime and $n \in \mathbb{N}$, one has to find an irreducible ...
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$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$

Prove that $$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$ I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right ...
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Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
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Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 ...
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Radical extension with root of cubic polynomial

If I take $f(x)$ is an irreducible cubic over $\mathbb{Q}$ with a root $\alpha$ in a splitting field and given that $\mathbb{Q}(\alpha)$ is a radical extension is it true that $\mathbb{Q}(\alpha) = ...
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Inertia group modulo $Q^2$

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
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If $a_1,a_2,a_3$ are roots $x^3+7x^2-8x+3,$ find the polynomial with roots $a_1^2,a_2^2,a_3^2$ [duplicate]

If $a_1,a_2,a_3$ are the roots of the cubic $x^3+7x^2-8x +3,$ find the cubic polynomial whose roots are: $a_1^2,a_2^2,a_3^2$ and the polynomial whose roots are $\frac{1}{a_1}, \frac{1}{a_2}, ...
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Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
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Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $

I'm trying to solve the following problem: Let $ L =\mathbb{Q}(\varepsilon_5) $ be an extension of $ \mathbb{Q} $. Find the Galois group $ G(L / \mathbb{Q})$ $ \varepsilon_5 $ is the primitive root ...
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Decomposing the ring of integers of a number field as a sum of prime plus field fixed by ramification group

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
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How to find the Galois group of a given polynomial from Galois viewpoint?

Excerpt from Page 3 Consider the polynomial equation \begin{equation*} x^4 - 2 = 0, \end{equation*} with integer coefficients. We have four solutions: \begin{equation*} \alpha = ...
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Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
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Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$

Let $\mathbb{K}=\mathbb{Q}(\sqrt[10]2)$. Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$. Well, it is easy to see that the degree of this extension over $\mathbb{Q}$ is ten. Also, is ...
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What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
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Roots of $f(x) = x^3+x^2-2x-1$

Roots of $f(x) = x^3+x^2-2x-1$ Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is ...
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Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
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On the expression of the Galois conjugates in terms of the coordinates in a basis

Let $K$ be a field and let $L$ be a Galois extension of $K$. Assume that $[L:K]=n$, and consider $e=(e_1, e_2, ...,e_n)$ a basis of $L$ over $K$. We note ...
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How to find minimal polynomial of primitive element (field theory)

I am given a primitive element $\alpha$ in the Galoisfield $F_{2^6}$. The question is to find the mimimal polynomial of $\alpha^7$. How to I find this? My thoughts so far: $$ \alpha^7 \rightarrow ...
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Irreducible Polynomials, and Galois Groups

I have a small question about some concepts. I know that if I have a polynomial $p(x)$ over a field $K$ and an extension field $F$, that elements of $\text{Gal}(F/K)$ permute the roots of $p(x)$ ...
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Why are inertia and decomposition groups only defined over normal extensions?

Can't we define them as the subgroups of the automorphism group of an arbitrary extension $L/K$ that fixes a prime $Q$ in $\mathcal O_L$ over a prime $P$ in $\mathcal O_K$? An ideal answer would ...
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What is the gcd of $x^3-2$ and $x^2+3ax+3a^2$?

I'm trying to understand Galois theory. $f(x)= x^3-2$, with $a$, $b$ and $c$ as roots. Galois resolvent $V_0= a+2b+3c$, $V_1= c+2a+3b$,... etc till $V_5=a+2c+3b$ assume $X=V_0$ ...
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Splitting field and galois group of $x^4-2x^2-1$ over $\Bbb{Q}$

I have that the roots of $m(x)=x^4-2x^2-1$ are plus minus $a=\sqrt{1+\sqrt{2}}$ and $b=\sqrt{1-\sqrt{2}}$ which is complex because $1-\sqrt{2}$ is negative. Taking out $i$ and letting $b=ic$ we have ...
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Extension of fields $[L:K]=2$. Show that there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$.

Let $K$ be a field with characteristic different from two, then if a extension of fields $L/K$ is such that $[L:K]=2$ then there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$. ...
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Isomorphism type of the Galois group

$f=(x^2-2)(x^3-3)$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. a) Determine the degree of extension of $K$ over $\mathbb{Q}$. b) Determine the isomorphism type of the Galois group of $K$ ...
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Showing polynomial is irreducible over field containing roots of unity.

Given a field $F$ containing all the roots of unity I'm trying to show that $f(x) = x^p - \alpha^p$ is irreducible over $F$ (where $\alpha$ is not in $F$). It's clear that $f$ splits in $F(\alpha)$ ...
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An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
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Determine the galois group of a quartic

I'm reading Hungerford's algebra chapter about galois theory. There is the following theorem in p.273 (with some minor changes) about determining the Galois group of a quartic: Let $K$ be a field ...
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Galois group of $(T^4-3)(T^6-3)$

Given the polynomial $f(T) = (T^4-3)(T^6-3)$, I would like to calculate the Galois group of $f$. What I've done is the following: setting $\alpha = 3^{1/4}$ and $\beta= 3^{1/6}$, $\zeta_k = e^{2\pi i ...
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Relations between galois group of polynomial and its factors

$f = f_1\dots f_n$ where $f_i$ is irreducible and distinct. What can i say about $\operatorname{Gal}(f/\mathbb{Q})$ if i know $\operatorname{Gal}(f_i/\mathbb{Q})$ for any(some) $i$.
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Galois group of $x^6-5x^3+6$

Let $f = x^6-5x^3+6$. I want to determine $\operatorname{Gal}(f/\mathbb{Q})$ without some group theory tricks (like Sylow's theorems) and without reduction $\bmod p$. Let $L_f = ...
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Galois group of $x^6-9$

$f = x^6-9 = (x^3-3)(x^3+3)$ Let $L_f$ be splitting field therefore $L_f = \mathbb{Q}[\sqrt[3]{3},e^{\frac{2\pi i}{3}}]$, $[L_f:\mathbb{Q}] = 9$. Also $Gal\space x^3±3/\mathbb{Q} = S_3$ and $Gal\space ...
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Prove the form of a fixed field

let $C_3=\langle\sigma\rangle$ and let $\sigma$ act on $K(s,t)$ by $s \mapsto t$ and $t \mapsto -s-t$ I want to prove that $K(s,t)^{C_3}=K(u,v)$ with $u=\frac{s^2+t^2+st}{st(s+t)}$ and ...
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Compositum of normal extensions is a normal extension

I'm trying to prove that if $ F \subset K, F \subset M $ are normal extensions, $ K,M \subset E $, then $ KM$ is also a normal extension of $ F $. I tried using the fact that $ F \subset KM $ is a ...
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Determine the Galois group of $ F(x^5) \subset F(x) $

I'm rather new to Galois theory and have been given this exercise: Suppose $ F $ is respectively equal to $ \mathbb{Q}, \mathbb{C}, \mathbb{F}_5 $ (the third one is just the 5-element field). My task ...
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Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
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Summary of Galois quintic unsolvability

Is this a sufficient flow of logic? Summary: Consider a fifth degree polynomial that has a formulaic solution. Then we have a radical extension with a string of subgroups of the Galois group which ...
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Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is an algebraic and infinite extension of Q

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is a algebraic and infinite extension on Q. Well, if i consider for every p prime, the polynomial p(x)=x^2−p, then p(x) is in Q(p√∣p is ...
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Galois of $x^3+2$

the splitting field of $x^3+2$ is $E=\mathbb Q(\sqrt[3]{2},i\sqrt{3}$) and I know the $Gal(E:\mathbb Q)$ has six elements which is isomorphic to $S_3$. How to find all intermediate subfields using ...
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Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
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Galois theory question (Kummer's Theory)

This question is from a past exam paper. Suppose $E = \mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3} - 39X + 26 \in \mathbb{Q}[X]$. Prove that $E$ is cyclic Galois extension of ...
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Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...