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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Compute Galois group of these extensions.

I think that I have a problem in the redaction. I have to compute 1) $G=\text{Gal}(\mathbb Q(\sqrt 3,\sqrt 2)/\mathbb Q)$ I know that $\mathbb Q(\sqrt 3,\sqrt 2)$ is the splitting field of ...
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1answer
31 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
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Calculating $Aut(\mathbb{Q}(\sqrt[3]{2}/\mathbb{Q}(\zeta))$

I am considering the Galois extension $L/\mathbb{Q}$, where $L = \mathbb{Q}(\sqrt[3]{2}, \zeta)$, and $\zeta$ is a primitive cube root of unity. I've found that $Aut(L/\mathbb{Q})$ can be viewed as ...
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Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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1answer
27 views

Intermediate field extensions and Degree of field extension

I was wondering whether there is a relation between $[L:K]$ and the number of intermediate fields $F$, where $K\subseteq F\subseteq L$. If there is then can someone please explain why. What if $L/K$ ...
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1answer
25 views

Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, ...
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Finding fixed subfield of $\mathbb{Q}(t)$

I'm looking at automorphisms $t \to 1-t$ and $t \to \frac{1}{t}$ of the field $\mathbb{Q}(t)$. By looking at the relations between these I think I've found the group generated by them to be $S_3$. ...
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1answer
10 views

Compositum of all Galois extensions of prime power degree

Let $\mathbb{K}$ a field of characteristic $\not =p$ prime, and $\mathbb{K}_p$ the compositum of all the Galois extensions of $\mathbb{K}$ whose degree is a power of $p$. I have to prove that for ...
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1answer
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Question on Galois

1) If $E/F$ is a Galois extension and $B$ is an intermediate field, then $E/B$ is a Galois extension. If $E/F$ is a Galois extension, then $E$ is a splitting field of some $f(x)\in F[X]$. Can I just ...
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1answer
23 views

Basis for field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$

I'm trying to find a basis for the field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$, where $\zeta$ is the cube root of unity. I attempted this with starting with a set of elements I know ...
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1answer
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Galois group of $f = x^5 - 2x^3 - x^2 + 2$

I've made some progress finding the Galois group of $f = x^5 - 2x^3 - x^2 + 2$ but I am having some difficulties. I've factorised it over $\mathbb{Q}$ as $f = (x^2 - 2)(x-1)(x^2 + x + 1)$ and so I ...
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1answer
29 views

Finding whether a field extension is normal and/or separable

I have the field extension $\frac{\mathbb{Q}(\sqrt[4]{-7})}{\mathbb{Q}(\sqrt{-7})}$ And i have to show whether it is normal and whether it is separable. I know that this extension is both separable ...
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1answer
26 views

Galois group of a polynomial and subfields

Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible quartic whose splitting field has Galois group $S_4$ over $\mathbb{Q}$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$. a) Prove ...
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2answers
51 views

Showing this field extension is not normal

This question is part of a homework assignment. We are asked if the field extension $\mathbb{Q(\sqrt[4]{-7})}/\mathbb{Q}$ is normal. Here is what I have so far: The obvious thing to do seems to be ...
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Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
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1answer
41 views

Converse of Galois Theorem.

If $F\subset K\subset E$ field extension such that $K\subset E$ and $F\subset K$ is both finite and Galois extensions then $F\subset E$ is Galois extension. My intuition says that this is false but I ...
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Let $f = x^4 + ax^2 + bx + c \in K[x]$ be an irreducible and separable polynomial and $L/K$ be its splitting field extension.

Let $disc(f) = D = \Delta^2$ be denoted as the discriminant of $f$ and assume that $\Delta \notin K.$ Let $g \in K[x]$ be the resolvent cubic of $f$ and assume there exists only one root which is $d ...
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question on Galois group

Let $f(x)\in F[x]$, let $E/F$ be a splitting field, and let $G=\text{Gal}(E/F)$ be the Galois group. 1) If $f(x)$ is irreducible, then $G$ acts transitively on the set of all roots of $f(x)$ (if ...
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1answer
24 views

$F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
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Splitting field of $1 + x^3 + x^6 + x^9$

I'm studying for a qualifying exam and wanted to know if my reasoning on this problem was correct: Let $L$ be the splitting field of $f(x) = 1 + x^3 + x^6 + x^9$ over $\mathbb{Q}$. Find the Galois ...
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1answer
51 views

On Galois closure

I'm working on this problem in Hungerford: For $\sigma \in Aut_F \bar{F}$, show that every finite extension of $K$, the fixed field of $\sigma$, is cyclic. For a finite extension $L$ of $K$, let $M$ ...
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1answer
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degree of extension of rationals adjoin infinitely many roots

Consider a radical extension of $\mathbb{Q}$ in which infinitely many numbers are adjoined. Each of these numbers is the $n^{th}$ root of some integer, where $n$ can vary. Can any information be ...
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Normal basis in every subgroup

Let $K$ be finite Galois extension of $k$, $G=Gal(K/k)$ and $k$ infinite. Prove: Exists $\beta\in K$ such that for every $N<G$, we have $\{\sigma(\beta):\ \sigma\in N\}$ is a basis for ...
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Is the Galois condition is necessary for the definition of cyclic extension?

Hungerford says that $F/K$ is cyclic, if $F/K$ is algebraic Galois and $Aut_K F$ is cyclic. Is it necessary that $F/K$ is Galois? Is there an example of extension which is cyclic, but not Galois?
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Cyclotomic polynomial identity [duplicate]

STATEMENT Show that $$\Phi_n(X)=\prod_{d\mid n}(X^{n/d}-1)^{\mu(d)}$$ where $\Phi_n(X)$ is the n-th cyclotomic polynomial. QUESTION Could anyone offer any help on how to show this result.
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Galois group of $E/K$ & Galois group of the extension $E$ over Fixed field?

I've proved this result for myself, but I have doubt in my proof whether it is true : Let $E/K$ be a field extension and $G(E/K)$ its Galois group. Suppose $E^{G(E/K)}$ is its Fixed field, i.e. ...
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Example of degree 4 polynomial with Galois group $S_{4}$

I know the condition in terms of discriminant and resolvent cubic when Galois group of a degree $4$ polynomial over $\mathbb{Q}$ has Galois group $S_{4}$. But I am looking for an explicit example of ...
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Construct degree $n$ field extension with no intermediate field

I want to construct degree $n$ field extension with no intermediate field for each $n$. I know for any finite group $G$ there is a Galois extension $K/F$ so that $Gal(K/F)$ is $G$. So my idea was to ...
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1answer
27 views

Fixed fields,cyclic groups and Galois theory

Let $G=S \times T$ where $S$,$T$ are both finite cyclic groups. Question 1: Is it true that there exists a Galois finite extension $L/F$ such that $Gal(L/F) \cong S \times T$? (I can't recall if ...
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1answer
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$\{1,\sqrt [3]{7},{\sqrt [3]{7 }}^{2}\}$ is a basis for $Q[\sqrt {2},\sqrt {3},\sqrt [3]{7}]$ over $Q[\sqrt {2},\sqrt{3}]$

I am trying to prove that $\{ 1,\sqrt [ 3 ]{ 7 } ,{ \sqrt [ 3 ]{ 7 } }^{ 2 }\} \quad is\quad a\quad basis\quad for\quad Q[\sqrt { 2 } ,\sqrt { 3 } ,\sqrt [ 3 ]{ 7 } ]\quad over\quad Q[\sqrt { 2 } ...
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1answer
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Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...
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1answer
25 views

Showing two field extensions (of Q) are isomorphic using primitive elements, and why every element is primitive

I'm taking a class on Galois Theory in another language and the prof is saying my answer on this is incorrect and I'm wondering why, particularly since sometimes there's a language barrier. Basically ...
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Aut$(K/F)$ permutes roots of polynomial.

Let Aut$(K/F)$ is the set of all automorphism from $F$ to $K$, where $K$ is a galois extension of $F$. Let $f(x) \in F[x]$ and $\alpha$ be a root of the polynomial $f(x)$. I am able to prove that for ...
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1answer
61 views

Extending field homomorphisms to automorphisms

I have $L/K$ a finite field extension and an irreducible polynomial which has two roots in $L$, $\alpha$ and $\beta$. I'm trying to show there is an automorphism of $L$ that fixes $K$ and switches ...
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Subgroups and corresponding subfields of galois group of $x^6 + x^3 + 1$

I think I have found the correct Galois group of $f = x^6 + x^3 + 1$ over $\mathbb{Q}$ to be $C_6$, the splitting field being $\mathbb{Q}(w)$ where $w$ is a complex 9th root of unity. Now I am trying ...
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Solvability by radicals of Polynomials defined by a recurrence relation

I want to determine the smallest integer $m$ such that the polynomial $P_{n}(x)$, $n\geq m$, given by : $$\left \lbrace \begin{array}{l} P_{n+1}(x) = P_n(x) (x-n-1) + \prod\limits_{i = 0}^n x-i\\ ...
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Transcendental extensions

I have to solve an exercise on transcendental extensions: 1) Show that a field extension $F \subset K$ which has transcendence degree at least 2, cannot be simple. 2) Two purely transcendental ...
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1answer
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Calculating fixed fields

I have to show that $\mathbb{Q}(\zeta)^{<\sigma>}$ $=$ $\mathbb{Q}(\zeta + \frac{1}{\zeta})$ $\sigma \colon L \to L$ is defined by $\sigma(\alpha) = \overline{\alpha}$, where ...
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For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$

Well I attempted to try this but I failed to solve it: So $$\Phi_{p^{n}}(x)= \frac{x^{p^{n}} - 1}{\Phi_{1}(x) \Phi_{p^{2}}(x) \dots \Phi_{p^{n-1}}(x)}$$ Now I'm just stuck here. I saw a result ...
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Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...
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Automorphism(Galois groups) and galois theory

I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is: Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of ...
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Galois group is $S_{n}$

If $K/F$ is Galois extension with Galois group $S_{n}$ then show that $K$ is the splitting field of a degree $n$ polynomial irreducible over $F$. We know $K$ is splitting field of some separable ...
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1answer
44 views

Degree 4 extension of $\mathbb {Q}$ with no intermediate field

I am looking for a degree $4$ extension of $\mathbb {Q}$ with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible ...
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Krull's Topology

Let $f_i : Gal(L/K) \mapsto Gal(L_i/K)$ the restriction $\sigma \mapsto \sigma_i=\sigma |_{L_i}$ and $\mathfrak{L}=(L_i)$ the system of a finite intermediate Galois extensions of $K$. We know this ...
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About Galois Covering Theory

so I am studying somethings about Galois Covering and I am writing a beamer to present for my friends of the university. But I would like of somethings about the author of Covering Galois Theory to ...
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Order of conjugate map

Let $\sigma :L\rightarrow L$, s.t $\sigma (\alpha)=\bar{\alpha}$. I've been asked to show that $\sigma\in Aut(L/K)$(the set of all automorphisms for the field extension) has order 1 or 2. I'm ...
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1answer
25 views

Order of automorphism = Order of field extension property

I've read that if $L/K$ is a field extension and $|Aut(L/K)|=|L/K|$, then L/K is a galois extension. I was wondering whether the converse is true, i.e if $|Aut(L/K)|\neq |L/K|$, then can we just ...
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Are most rational quintics unsolvable?

It is well-known that, as polynomials of degree exceeding 4, there exist quintics whose roots cannot be solved for by radicals (Abel-Ruffini theorem). So we can divide the set of rational quintics ...
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1answer
25 views

Is Q(4th root(2))/Q a galois extension

I'm having some difficulty with the definition for galois extensions. The definition as read from my notes is $L/K$ is galois if $L^{Aut(L/K)}= K$. Where $Aut(L/K)$ is definied to be the set of all ...
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1answer
44 views

Degree of splitting field less than n!

I've been asked to prove that if a function $f\in \mathbb{Q}$ has degree $n$, then the splitting field of $f$ has degree less than or equal to $n!$. That is $\mathbb{Q}(\alpha_1, ...