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 (galois-...

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Is this theorem provable using relatively elementary number theory and abstract algebra?

$\textbf{Theorem}$: Let $p$ be a prime. Let $q$ be a prime that doesn't divide $p - 1$, so that $\mathbb{F}_p$ does not have an element of order $q$. Let $\zeta$ be an imaginary number whose order is $...
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How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$?

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$? I think that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}):\mathbb{Q}] = 8$, but not really sure how to ...
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Galois group of the splitting field of $x^3-2$

I want to find $Gal(E/\mathbb{Q})$ where $E$ is the splitting field of $f(x)=x^3-2$. I started out finding the zeros, which is $2^{1/3},2^{1/3}\omega, 2^{1/3}\omega^2 $, where $\omega={-1+i\sqrt3\...
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Prove every finite extension of subfields of $\mathbb{C}$ is simple

This is a question is from a past paper, and it's only worth 8 marks, and it's really got me. I'm allowed to assume that if $L/K$ is such an extension, that there are $[L:K]$ $K$-embeddings of $L$ ...
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Characterizing the Galois group using permutations of roots

I'm studying Galois theory at the moment. It seems to me that the fundamental motivation for the Galois group is the following. We know that if a rational function of the roots is invariant under ...
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Why does this method of proof for $\overline{\mathbb{F}_{p}}$ not apply for $\mathbb{F}_{p}$

Let $\overline{\mathbb{F}_{p}}$ be the algebraic closure of the finite field $\mathbb{F}_{p}$. Consider $F:= \overline{\mathbb{F}_{p}}(x,y)$ and let $K:=\overline{\mathbb{F}_{p}}(x^{p},y^{p})$. Below ...
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Splitting field of an irreducible polynomial of degree five $f \in \mathbb{Q}[X]$

Let $\Omega/\mathbb{Q}$ be the splitting field of an irreducible polynomial $f \in Q[X]$ of degree five. Show that $Gal(\Omega/\mathbb{Q})$ equals $A_5$ or $S_5$ if $Gal(\Omega/\mathbb{Q})$ has an ...
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640 views

So-called Artin-Schreier Extension

Let $F$ be a field of characteristic $p$. Let $K$ be a cyclic extension of $F$ of degree $p$. Prove that $K=F(\alpha)$ where $\alpha$ is a root of the polynomial $p(x) = x^{p} - x - a$ for $a \in \...
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Prove that every finite group occurs as the Galois group of a field extension of the form $F(x_{1}, \dots , x_{n})/F$.

I've seen that every abelian group is a Galois group over $\mathbb{Q}$ for some subfield of a cyclotomic field, but I'm not sure about this more general result.
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Simple group question need help…

Alright so I've got a question here in terms of groups. So define $\omega = {e}^{2i\pi\over 13}$ -The exponent of e should be $2i\pi\over 13$ but it's not coming clear when as an exponent of e there ...
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Splitting field of $f=X^p -a \in \mathbb{Q}[X]$.

Let $p$ be a prime number, and $a \in \mathbb{Q}$, a number such that there is no integer $k$ satisfying $p^k=a$. Write $f= X^p -a \in \mathbb{Q}[X]$. I have to prove the following statements: The ...
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Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
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Is is true that $\zeta$ has finite order?

Let $\zeta$ be a complex number on the unit circle $\{z\in \mathbb{C}: |z|=1\}$.Suppose that $[\mathbb{Q}(\zeta):\mathbb{Q}] < \infty$.Is it true that $\zeta ^n=1$ for some positive integer $n$?
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Degree of splitting field of $X^n-1$ over $F_p$

Suppose that $n$ is a natural number and p is a prime that does not divide n. Let $L$ be the splitting field of the polynomial $X^n-1$ over $\mathbb{F}_p$. Show that $[L:\mathbb{F}_p]$ is the smallest ...
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Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. Further,...
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Degree of non-separable extension

Suppose $K$ is a field with charasteristic number $p$. Further suppose that L is a non-separable extension of K with $[L:K]=k$. Why does it hold that k is a multiple of p?
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Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
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Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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does the polynomial split?

Let $a \in \mathbb{C}$ such that $a^n $ is rational for some positive $n$.Let $ m$ be the smallest positive integer satisfying previous condition.Is it true that $ x^m- a^m$ splits in $\mathbb{Q}(a)[x]...
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Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
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Integral closures and Galois extensions

I was reading in Lang's Algebraic Number Theory (Second Edition page 15-16) and the following proposition occured. Proposition 14. Let $A$ be integrally closed in its quotient field $K$, and let $B$ ...
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what is the minimal polynomial of $\alpha=\rho+\rho^4+\rho^{16}$ , $\rho^{21}=1$?

here is a question that I recently tried to solve: Is it possible to construct with Compass-and-straightedge the number $$\alpha=\rho+\rho^4+\rho^{16}$$ , while $\rho^{21}=1$ over $\mathbb{Q}$ ? also ...
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Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the $[K_i:K_{i+1}]$...
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77 views

The Galois closure

If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
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Proving the Galois Group of an extension is abelian

Let $E_{1}, E_{2}$ be subfields of $\mathbb{C}$. Suppose $E_{1}|\mathbb{Q}$ and $E_{2}|\mathbb{Q}$ are finite Galois extensions and $G(E_{1}:\mathbb{Q})\cong$ $\mathbb{Z}_{6}\cong$ $G(E_{2}:\mathbb{Q})...
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Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
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Proof Verification of the Primitive Element Theorem

Hypothesis: Let $E$ be a two-dimensional separable extension of $F$. Say $E = F[\alpha, \beta]$. Then $\exists \gamma \in E$ s.t. $E = F[\gamma]$. Note that the case for beyond two dimensions ...
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Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
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Isomorphism betwixt a Galois Group and the set of $n$th roots of unity

Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...
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A Question about Cubic and Galois Fields

Let $f \in \mathbb{Q}[x]: f(x) = x^3 + p x + q, p \geq 0, f$ irreducible therein; $\mathbb{k}$ be the splitting field of $f$ over $\mathbb{Q}$; $G = Gal(\mathbb{k} / \mathbb{Q})$. I have to show that ...
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Roots of Unity: Sums, Products, and Field Extensions

(1) I have to prove the following: $\forall p \in \mathbb{P} \setminus \left\{ {2}\right\}, \sum_{k=1}^{p-1}(\zeta_{p}^{k})=-1$ where $\mathbb{P}=\left\{ {p\in \mathbb{Z}:p>0, prime}\right\}$. ...
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$F \subset L \subset E$ s.t. $L,E$ Normal $\implies$ Elements of $G(E/F)$ Map $L$ onto itself.

Problem: Let $E$ be a finite, separable normal extension of $F$. Suppose that $F \subset L \subset E$ s.t. $L$ is also normal over $F$. Prove that the elements of $G(E/F)$ map $L$ onto itself. ...
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$p(x) \in F[x]$ Has One Root in Normal Field Extension $E$ $\implies$ $p(x)$ Splits Entirely in $E$

Problem: Suppose that $E$ is normal over $F$. Let $p(x) \in F[x]$ have one root in $E$. Show that $p(x)$ therefore splits entirely in $E$. EDIT: As per the comment below, assume also $p(x)$ is ...
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The degree of the splitting field of $X^6+X^3+1$

Suppose $L \subset \mathbb{C}$ is a splitting field for the polynomial $X^6+X^3+1$ over $\mathbb{Q}$. Determine $[L:\mathbb{Q}]$ I have several solutions for the problem. However I'm having trouble ...
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Understanding the Fundamental Theorem of Galois Theory (Artin's Text)

Note: I'm using Emil Artin's free text on Galois Theory to understand the Fundamental Theorem of Galois Theory. Hypothesis: Let $F \subset E$ s.t. $E$ is the splitting field for separable $p(x) \in F[...
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What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p $ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
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find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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Determine $\text{Gal}(\mathbb{Q}(\alpha) / \mathbb{Q})$ where $\alpha = \omega + \omega^7 + \omega^{11}$

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. Determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. With which other well ...
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Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong \...
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If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?

Defn: A field extension $K/F$ is obtained by adjoining $n$th roots if there is a tower of fields $F=K_1\subset\cdots\subset K_n=K$ such that for each $i$, $K_{i+1}=K_i(\alpha_i)$ and there exists some ...
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Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
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Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
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Galois Theory: Finding the discriminant of a polynomial

I am very confused on how to determine the coefficient of: $f(x)=x^3-s_1x^2+s_2x-s_3$. Artin is really proving me with too much abstraction, is there an easier way to be thinking about how to do this. ...
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Lost on Galois Group over $\mathbb{Q}(i)$ Question.

The goal is to find the Galois Group of $t^8-i$ over $\mathbb{Q}(i)$ where $i=\sqrt{-1}$. I've found the roots of the polynomial because I know I'll need them and they are $\pm i^\frac{1}{8}$,$\pm ...
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293 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ \mathbb{1}...
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317 views

Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
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Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
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380 views

Every Intermediate Field of Abelian Galois Field Extension is Splitting Field of a Separable Polynomial

This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
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Construct an embedding

I'm dealing with this problem from the book "Field Theory" (Steven Roman) Suppose $F$ and $E$ are fields and $\sigma : F \rightarrow E $ is an embedding. Construct an extension of $F$ that is ...