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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Extension Field of $\mathbb{Q}$ and its Galois group

How many elements are in $Gal(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}:\mathbb{Q})$?
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Subgroups of Galois group and intermediate fields lattice for $(x^3-2)(x^2-3)$

I am trying to systematically determine all subgroups of Galois group and intermediate fields for $(x^3-2)(x^2-3)$(over $\mathbb Q$). It's not hard to determine the Galois group of $(x^3-2)$ and ...
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Galois extension of $\mathbb Q$ of degree $n$

I have a basic question in Galois theory. For any given natural number $n$ is there a Galois extension of $\mathbb Q$ of degree $n$? I want to show that there are splitting fields of polynomials in ...
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About the Galois group of $\mathbb{Q}[\sqrt{5},\sqrt{2+\operatorname{i}}]/\mathbb{Q}[\sqrt{5}]$

Some notations: let $K=\mathbb{Q}[\sqrt{5}]$, $N=\mathbb{Q}[\sqrt{5},\sqrt{2+\operatorname{i}}]$, $a=\sqrt{2+i}$, $b=-a$, $c=\sqrt{2-i}$, $d=-c$. I know that $N/K$ is normal because it decompose the ...
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separable polynomial $ \bmod p$ (definition)

Given a polynomial $ f(x) \in K[x]$, where $K$ is a number field, we say that $f$ is separable if all its roots are distinct in an algebraic closure of $K$. Question: What does it mean a ...
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Transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements.

I need to show that transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements. Since I do not know much about ordinals and cardinals, a proof based on algebra (rather than ...
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Solving polynomial equations using more than radicals

It is well known that one cannot solve every polynomial equation over $\Bbb Q$ using just radicals. In other words, let $A_n = \{x^n - a\mid a \in \Bbb Q\}$, $A = \cup_n A_n$ and $\bar{\Bbb Q}$ the ...
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Showing $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$

I want to show that $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$, but I'm not that familiar with computing Galois groups so I don't really know how to do this exercise. How do I approach ...
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(Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$

(Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$. This is my rough proof to this question. I was wondering if anybody ...
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Whether an embedding is an automorphism

Let $K/F$ be a field extension and let $\sigma$ be an embedding from $K$ into $K$ over $F$. If $K/F$ is algebraic, prove that $\sigma \in Aut(K)$. I know how to prove the case when $K/F$ is finite. ...
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Galois group of the simple extension

Let $K=Q(\sqrt3,\sqrt5)$. Show that the extension is $K/Q$ is simple and also Galois extension. Determine its Galois group. I showed that extension is simple because $K=Q(\sqrt{3}+\sqrt{5})$ But I ...
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Field extension of fixed field has degree greater than the size of the group

Let $K$ be a field, $G\leqslant\mathrm{Aut}(K)$ a (finite) group, and $K^G$ the fixed subfield of $K$. How exactly would you go about proving the following?$$[K:K^G]\geqslant|G|$$ For some reason I ...
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Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
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Galois Theory and Splitting Fields

So I have an exam tomorrow and I think I'm rather prepared as far as the theory goes (I have the theorems in the book memorized, etc), but I am rather worried about any "concrete" questions I may get ...
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Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof ...
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Whether a field extension contains $i$.

For which values of n does the cyclotomic extension over $Q$ contain $i$? My guess is that this is precisely when n is divisible by 4. If n is divisible by 4,then i can show this quite easily. But is ...
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Whether or not an extension is Galois over $\mathbb Q$

Is the extension $\mathbb Q(\sin\frac{2\pi}n)$ a Galois extension over $\mathbb Q$? For the case when $n$ is divisible by $4$, I know that this happens. But I don't know how to do this in general. ...
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Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...
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Compute the splitting field and the Galois group of $x^4 - 5$ over $\mathbb{Q} (\sqrt{5})$. [duplicate]

I believe the splitting field is easily found by the following. $x^4 - 5 = (x^2 - \sqrt{5})(x^2 + \sqrt{5})$, so the splitting field is $\mathbb{Q}(\sqrt[4]{5},i)$, or is this incorrect? Once I ...
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Weakly normal polynomials and normal polynomials.

I have been going through the notes of Prof. Pete L. Clark here (warning: long pdf). They are rough notes on Field theory and on page 30 he defines $P \in K[t]$ a normal polynomial if $P \in L[t]$ is ...
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Tower within a Galois extension

Consider the following tower of fields: $$ K \subset M \subset L $$ If $ L/K$ is a finite Galois extension, then is it true that $ M/K $ is a Galois extension ? Is it also finite ? It is clear to ...
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Let $F/k$ be a Galois extension. Show there exists an $\alpha \in F$ such that $\{ \sigma(\alpha) | \sigma \in G (F/k)\}$ is a basis for $F$ over $k$.

Title. I need to show that $\{ \sigma(\alpha) \mid \sigma \in G (F/k)\}$ forms a vector space for $F$ over $k$ if $F/k$ is a Galois extension. I know that if $F/k$ is Galois, then $F$ is the ...
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Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
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Automorphism of $\mathbb{Q}({\zeta_n})/\mathbb{Q}$

I came across the theorem where, for $n=p^{a_1}\cdots p^{a_m}$: $Gal(\mathbb{Q}({\zeta_n})/\mathbb{Q})\simeq$ $Gal(\mathbb{Q}({\zeta_{p^{a_1}}})/\mathbb{Q})\times ...
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What are all the intermediate fields of $\mathbb{Q}\big(\sqrt{3+\sqrt{5}}\big)$ containing $\mathbb{Q}$?

I've come to a fork in the road, and it is sending me on wild goose chases. This question comes from a final exam for an Intermediate Abstract Algebra course I just took this past Spring. I'm ...
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Frobenius automorphism and cyclotomic extension

There is a lemma from a lecture I attended where I scribbled down notes and try to make sense of the proof afterwards and there is a spot at which I am stuck. First I have to set up some notations and ...
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Isomorphisms in characterisation of Galois extension

My definition of an extension $M/K$ to be Galois is that $Gal(M/K)$ only fixes things in K. I'm trying to prove that this is equivalent to $M/K$ being normal and separable. I know that fact that if ...
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Why is $Gal(\Omega/k)$ a topological group under the Krull topology?

For an infinite Galois extension $\Omega/k$ the Krull topology on $G:=Gal(\Omega/k)$ is defined by taking as a basis for the neighbourhood of an element $\sigma \in G$ all cosets of the form $\sigma ...
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Constructing a tower to the splitting field of an arbitrary irreducible cubic.

Let f(x) = x3 + ax2 + bx + c ∈ $\mathbb{Q}$[x]. Let K be the splitting field of f(x). I want to construct a tower: $\mathbb{Q}$ ⊂ K1 ⊂ ... ⊂ Kr = K Each Ki = Ki-1(α) where either ...
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Galois group of the field of all constructible complex numbers

I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call ...
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Compute Gal($\mathbb{Q}(w^{k})/\mathbb{Q})$ up to isomorphism, for all k $\in \mathbb{Z}$, where $w=e^\frac{2\pi i }{10}$

Compute Gal($\mathbb{Q}(w^{k})/\mathbb{Q})$ up to isomorphism, for all k $\in \mathbb{Z}$, where $w=e^\frac{2\pi i }{10}$ So I have got for k=1 That $\mathbb{Q}(w) $ contains 9 "roots" $w, w^2, ...
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Galois Extension whose Galois Group is $\mathbb{Z}_2\oplus\mathbb{Z}_4$

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Let $E/F$ be a Galois extension with Galois group isomorphic to ...
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“Implicit” condition about separability of a quartic polynomial

Here is an exercise in Hungerford's Algebra, page 277 Ex.12. Let $K$ be a subfield of real numbers and $f \in K[x]$ an irreducible quartic polynomial(of degree 4). If $f$ has exactly two roots, the ...
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Showing explicitly that $\alpha \mapsto \alpha+1$ is an automorphism in the Galois group of $x^p-x+a$?

I have already shown that $x^p-x+a$ is irreducible and separable over $\mathbb{F}_p$ and I have shown that if $\alpha$ is a root, so is $\alpha+1$, so $\alpha$ generates the extension, i.e. the ...
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Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...
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Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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Galois groups and free products

This question comes from trying to better understand the generalisation of the following past exam question: The interesting case here is $G_2$: we see that the Galois group of $x^2+1$ is $C_2$ ...
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Showing a certain cyclotomic polynomial must split

This is part of a question from a book, I'll post the rest if anyone would like me to, but the bit I'm stuck with is: Suppose $\deg(L/K) = p$, a prime not equal to $\operatorname{char}K$, and $L$ is ...
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Show that f is irreducible over $\Bbb{F}_4(t)$, $t$ transcendental over $\Bbb{F}_4$.

I am trying to show that $f=x^9-t$ is irreducible in $K[x]$, where $K=\Bbb F_4(t)$ with $t$ transcendental over $\Bbb{F}_4$. Can someone give me a hint? Thanks. How do we determine the degree of the ...
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Galois Group of a Splitting field of $x^8-4$

Let $L$ be the splitting field of $f(x)=x^8-4$, I know that $|Gal(L/\Bbb Q)|=8$, but I have no idea what is the next step.
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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Galois group of $x^p-x-a$

$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$. First I showed that this is an irreducible polynomial and has no multiple roots. ...
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Permuting roots in splitting fields

Currently, I've just started to study Field and Galois theory. In one of my textbooks, I have found the following (probably important) theorem: If $K/F$ is a splitting field for the irreducible ...
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Image of archimedean place of a number field in $\mathbb C$

Let $L/K$ be a finite Galois extension of number fields and let $\phi$ be an embedding of $K$ into $\mathbb C$. Let $\psi_1$ and $\psi_2$ be two embeddings $L\to \mathbb C$ which extend $\phi$. ...
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Proving that extension by radicals implies solvable group

I'm trying to understand the following excerpt from Fraleigh's A First Course In Abstract Algebra, Seventh Edition, pp. 472-473: 56.4 Theorem Let $F$ be a field of characteristic zero, and let ...
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Finding the size of a Galois Group of a splitting field for a polynomial of degree 6.

Let $f(x) = x^6 + ax^4 + bx^2 + c$ with a,b,c ∈ $\mathbb{Q}$ be an irreducible polynomial in $\mathbb{Q}$[x]. Let K be the splitting field of f(x) over $\mathbb{Q}$ and let G = ...
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Primitive element Theorem without Galois group.

I want to know if exists a demonstration of the Primitive Element Theorem without using the Galois Group of the extension. Anyone knows a demonstration without it?
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Galois group of $x^4-1$ over $\mathbb{Q}$.

Let $p$ be the polynomial $x^4-1$. What is $\text{Gal}(p/\mathbb{Q})$? The splitting field is $\;\mathbb{Q}[i]\;$. The order of the Galois group is equal to the degree of the splitting field of ...
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Prove $f(x)$ is irreducible and its roots are real.

Let $f(x)\in\mathbb Q[x]$ be a polynomial of degree $3$ and let $E$ be the splitting field of $f(x)$ over $\mathbb Q$. Show that if $G(E/\mathbb Q)$ is isomorphic to $\mathbb Z/\mathbb 3\mathbb Z$, ...
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Galois group of $x^4-5$ over $\mathbb{Q}$.

I factored $x^4-5$ into $(x-\sqrt[4]{5})(x+\sqrt[4]{5})(x-i\sqrt[4]{5})(x+i\sqrt[4]{5})$, determining the roots: $x=\pm\sqrt[4]{5}$ and $x=\pm i\sqrt[4]{5}$. The $\mathbb{Q}$-automorphisms of $x^4-5$ ...