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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Specific question on imaginary quadratic field [closed]

How to solve the following question?! Let $K$ be an imaginary, quadratic field and let $L/K$ be a Galois extension. If $\tau$ is complex conjugation, show that: (a) $L/\Bbb Q$ is Galois iff ...
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Galois group of splitting field of a polynomial

Let $f$ be a polynomial with integer coefficients and irreducible over $\Bbb{Q}$. Let $p$ be a prime. Suppose $f(mod $ $p$) can be written down as a profuct of $r$ distict irreducible polynomials ...
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A $p^n$th root of unity is in $N_{F/k}(F^\times)$ if and only if there is an extension of $F$ which is a cyclic extension of $k$ of degree $p^{n+r}$

Hello, I have to solve this homework but I have completely no idea. please help me, it is very difficult for me. any attempt will be welcomed and appreciated. Conditions : $n, r$ are fixed number ...
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Cyclotomic extension contained in a sequence of Kummer extensions

Let $K$ be a field of characteristic $0$ and $L$ the $n$-th cyclotomic extension of $K$. Show that there is a sequence of Kummer extensions $E_0 = K \subseteq E_1 \subseteq \cdots \subseteq E_r$ ...
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Galois extension, simple extension, power

Hello, let $k$ be a field with characteristec zero. and suppose $k$($\alpha$) is Galois extension of $k$. then, is it true $k$($\alpha^3$) = $k$($\alpha$) ? or not? Thank you for your help.
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A simpler proof of an important statement in the book “Galois Theory” by Ian Stewart

Reading the book "Galois Theory" (3rd ed.) by Ian Stewart, I try for better understanding and possible future formalization to rewrite Stewart's proof of a very important Lemma 15.7 If $K$ is a ...
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A book for advanced field theory

I am searching for an alternative text to chapter 5 of Bourbaki for field theory, that covers, for example, separable and inseparable degrees. I know the basics about field theory and Galois theory. ...
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Galois group of $x^5 + x^2 + 1$ over the field $\mathbb{F}_{2}$

I'm uncertain if I am doing this problem correctly. This is an old algebra prelim problem regarding Galois theory. We have to find the Galois group of the irreducible polynomial (the problem already ...
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reducing depressed quartic to cubic polynomial

Recently I found this lovely algebraic equation solver flowchart online. What I do not understand is how to get from the Depressed Quartic (top right) to the Monic Cubic polynomial (the path where ...
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135 views

How to find algebraic connections between zeros of a polynomial?

Let $f(x)$ be an irreducible integer polynomial of degree $k$. Let $x_1,x_2,...,x_j$ be some zeros of $f(x)=0$ where $j<k$. How do I find identities of type $P(x_1,x_2,...,x_j) = 0$ where $P$ is ...
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66 views

Degree of an irreducible polynomial over a field which has a Galois group isomorphic to $ Q_8 $

Let $Q_8$ be the 8-element quaternion group. What is the minimum degree of an irreducible polynomial over a field which has a Galois group isomorphic to the $ Q_8 $ group?
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Determine the subgroup of the Galois group of $\mathbb Q[i\sqrt 3]$ and $E$ the splitting field of $(t^3-2)(t^3-5)$

Determine the subgroup $\text{Gal}(E/\mathbb Q[i\sqrt 3])$ where $E$ the splitting field of $(t^3-2)(t^3-5)$ and the intermediate fields. We clearly have that $E=\mathbb ...
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Finding a Galois group using a cubic resolvent

I've heard there's a fast way to find the Galois group of a quartic polynomial using its resolvent. Can anyone explain how it's done or give a reference? Is this method the same as the general ...
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Dummit and Foote page 591

Having trouble with something that Dummit and Foote is saying. On page 591 it says "Note that over $\mathbb{Q}$ or over a finite field (or, more generally, over any perfect field) the splitting field ...
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What are the intermediate fields of $\mathbb Q(\sqrt[3]2,e^{\frac{2i\pi}{3}})$ (Galois group)

The elements of Galois group are \begin{align*} \sigma _1:\mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}]&\longrightarrow \mathbb Q[\sqrt[3]2,e^{\frac{2i\pi}{3}}],\\ \sqrt[3]{2}&\longmapsto ...
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Galois group of $x^8-2$ and intermediate fields

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to ...
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Algebraic number of degree four that cannot be constructed with ruler and compass

The real number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$, where \begin{equation*} ...
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Finding Galois group of $x^6 - 3x^3 + 2$

I'm trying to find the Galois group of $$f(x)= x^6 - 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 - 2)$$ I can see the splitting field must be ...
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Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= ...
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The split closure of a radical extension is a radical extension

I'm stuck. I am trying to prove a lemma, it's from the Kaplansky's "Fields and Rings", and there's no proof. It says: LEMMA. If L is a radical extension of $K$ and $M$ is a split closure of $L$ ...
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Irreducibility of a polynomial with algebraically independent coefficients

I am learning some kind of field theory. Let $\mathbb{Q}'$ be the smallest subfield in $\mathbb{C}$ containing all roots of unity. Recently I read a book on Galois theory and met the following ...
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Galois finite extension

Let $K/ \mathbb{Q}$ be a finite Galois extension, $K \otimes_{\mathbb{Q}} \mathbb{R} \simeq \mathbb{R}^s \oplus \mathbb{C}^t$. Prove that either $s=0$ or $t=0$.
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Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
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When is a number square in Galois field p^n if it's not square mod p?

Here is the problem, that I'm stuck on. There is no square root of $a$ in $\mathbb{Z}_p$. Is there square root of $a$ in $GF(p^n)$? Well, it's certainly true that $$x^{p^n}=x$$ and $$x^{p^n-1}=1$$ ...
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A question on the relation of the Galois group as field automorphisms and the Galois group as permutations of roots

Let $f\in K[X]$ be a monic separable polynomial and $L$ a splitting field of $f$. Let $M=\{l_1,\ldots,l_n\}$ be the set of roots of $f$ in $L$, i.e. $$ f=(X-l_1)\cdots(X-l_n). $$ The Galois group ...
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Does the fixed field of automorphisms group characterize Galois extensions?

If $E/K$ is a field extension we use the notation $\def\Aut{\operatorname{Aut}}\Aut(E/K)$ for the set of field automorphisms of $E$ that are the identity over $K$. It's immediate that the set ...
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A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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When does a formula for the roots of a polynomial exist?

My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...
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Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I've also seen many applications of this fact ...
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Show that $E=\mathbb{Q}(a)$

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial with the splitting field $E$ and let the group $Gal(E/\mathbb{Q})$ be abelian. If $a$ is a root of $f(x)$ then $E=\mathbb{Q}(a)$. Could you ...
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Understanding the definition of solvable by radicals.

I am currently studying the third edition of Ian Stewart's book "Galois Theory". In the book, solvability of a polynomial by radicals is defined as follows: Let $f$ be a polynomial over a subfield ...
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Looking for GF(16), GF(32) … GF (256) tables

I'm learning about Galois Fields by implementing the code to create the addition/multiplication/log/ilog tables. I've got working code but I cannot find many of the actual Galois Field tables online ...
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How can I find the degree of the extension?

Let $\omega_7=e^{2\pi i/7}$ . How can I find the degree of the extension $\mathbb{Q} \leq \mathbb{Q}(\omega_7+\omega_7^5)$?? Could you give me some hints??
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Elements of $\mathbb{Q}(e)$

Is $e^n$ for $n$ an integer an element of the field $\mathbb{Q}(e)$?
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A question concerning cyclic field extensions.

In the study of cyclic extensions we have the following theorem: Theorem Let $K$ be a field containing an $n$-th primitive root of unity $\zeta$. Then the following claims hold: If ...
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Galois theory reference request

I ran into the following description of Galois theory in Gelfand and Manin's Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and ...
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Why do I get all the $K$-automorphisms of a splitting field $K''$ successively?

Please let me explain my question on a specific example: Let $K=\mathbb{Q}$ and let $K''=\mathbb{Q}(\sqrt[3]2, \xi)$ be the splitting field of the polynomial $f=X^3-2\in K[X]$. The polynomial $f$ is ...
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Galois group of $x^{15}-1$

Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity. a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$. b) Show that $G$ is isomorphic ...
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Find the field of intersection

Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. Could ...
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Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
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Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
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Finding the Galois Correspondence of polynomial $t^4-2$

For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$. I figured out that the Galois group of this polynomial is the dihedral ...
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Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
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Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
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Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is ...
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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 ...