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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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 $x^4-5$ ...
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Find the minimum polynomial of a sum of roots of unity.

Let $ \omega $ be an 11-th primitive root of 1 over $ \Bbb Q $ Let $ \beta = \omega + \omega^9 $ Find $ [ \Bbb Q ( \beta) : \Bbb Q ) ] $ and Find the minimum polynomail of $\beta$. I asked a ...
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Are these two fields the same?

I wanted to know if the field $\mathbb{Q}(i\sqrt{7}) = \mathbb{Q}(\sqrt{7}, i)$ are the same. I don't think they are because $i \notin \mathbb{Q}(i\sqrt{7})$?
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3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$

Taken from these notes [1] on Galois Theory, I would like to show that iterating the map $$p: x \mapsto x^2 - x - 2 $$ has a cycle of order 3 when you start with the root of $x^3 - 3x - 1 = 0$. ...
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prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
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Proving something is not a Normal Extension

Let $M = \mathbb{Q}(\sqrt{3}, i\sqrt[4]{5})$ be an extension of $\mathbb{Q}$. Then work out the basis of $M$ over $\mathbb{Q}$ and show that the extension $M/\mathbb{Q}$ is not a normal extension. So ...
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Explicit Galois Action for $X^3 - X -1$

I have always been frustrated with how indirect discussions of Galois Theory are in Algebra textbooks. Even in fine treatments such as Miles Reid. Are there any good examples where we can draw ...
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Question on Galois Theory [closed]

A finite normal extension K of a field F is cyclic over F is G(K/F) is a cyclic group. Show that if K is cyclic over F and E is a normal extension of F, where F≤E≤K, then E is cyclic over F and K is ...
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Galois group of $t^4-3t^2+4$

I am trying to understand all the steps for finding the Galois group of the extension $K:\mathbb{Q}$ where $K$ is known to be the splitting field over $\mathbb{Q}$ of $p(t) = t^4-3t^2+4$. We know that ...
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Homomorphisms preserve solubility of groups, and some others.

Definition: Let $G$ be a group. A subnormal series for G is a chain of subgroups $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq G_n = G$ such that $G_i$ is normal in $G_{i+1}$ for $i = 0,1, ..., ...
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Example of an non-normal inseparable field extension

I was asked to prove or disprove that if a field extension is not normal then it is separable. I can't see why this would be true so I want to disprove it with an example of a non-normal inseparable ...
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Trissecting by Ruler and Compass

I am learning Galois Theory by myself and studying its glorious applications. In the section of Straightedge and compass I got stuck at the following: Let be an angle whose cosine is equal to $1/9$. ...
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Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are ...
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Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...