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 Group of a Product

Q: Let K be the splitting field for $(x^{5}-1)(x^{3}-2)$ over $\mathbb{Q}$. Compute the cardinality of the Galois group $G$ for $\mathbb{Q} \subset K$, and show that G is not abelian. So first I ...
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Galois group solvable but $f$ not solvable.

I know from a theorem that: Let $F$ be a field of characteristic $0$ and $f(x)\in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is solvable. But what if the ...
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Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
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Is this a valid way to extend the proof of the insolubility of the quintic?

I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial ...
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Alternating groups as galois groups

Is there an elementary proof that the alternating group $A_n$, for any $n$, is the Galois group of an extension of the rationals? In fact, I am looking for a proof which does not use Hilbert's ...
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Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
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Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements ...
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having prime degree:being irreducible or having a root! [duplicate]

Let $p$ be a prime number. Prove that for any field $K$ and any $a \in K$, the polynomial $x^p−a$ is either irreducible, or has a root. it doesn't seem hard,but i have no idea. any hint is welcomed! ...
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Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over ...
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Unramified Galois extension of a function field of a curve - definition

Reading some papers I've encountered the following: Let $X$ be a smooth curve over a field $K$ with function field $K(X)$. Consider the maximal Galois extension of $K(X)$ which is unramified over $X$. ...
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Let $p$ be a prime such that Char$(K) \neq p$ and $x^p - t$ is irreducible in $K[x]$.

The question is: If $\omega$ be a primitive $p$th root of unity and $l = [K(\omega):K]$ and $L/K$ is the splitting field of $x^p - t$. Show that $\Gamma(L/K)$ can be generated by two elements ...
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Calculate the degree of $[\mathbb Q(a):\mathbb Q]$

How to determine $[\mathbb Q(a):\mathbb Q]$ for $a\in\overline{\mathbb Q}\setminus\mathbb Q$ with $a^p=1$ with $p$ prime? So $a$ is an algebraic, complex number which cannot be written as a quotient. ...
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How can you write $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ using a single algebraic element $\mathbb{Q}[\alpha]$?

Looking at the basis of $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ gives me no idea on how to generate it using $\{1, \alpha, \alpha^2,\alpha^3,\alpha^4,\alpha^5\}$ for some $\alpha$ algebraic over ...
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How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$?

1) How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$ where $K=\mathbb C(X)$ ? I would say that the minimal polynomial of $\sqrt[n]X$ on $K$ is $t^n-X\in K[t]$ which is separable, ...
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Calculating the degree of $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$

Let $m\in\mathbb Z$ with a prime factorization of the form $m=p\Pi p_i^{n_i}$, $p\neq p_i$. How can I calculate $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$ for a natural number $n$?
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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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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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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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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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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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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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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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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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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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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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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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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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$F/E$ 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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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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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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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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$\{1,\sqrt [3]{7},{\sqrt [3]{7 }}^{2}\}$ 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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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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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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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 ...