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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Map from Galois extension to its third tensor power

Suppose $L/K$ is a finite Galois extension of fields, with group $G$. Then $G$ acts on $\mathrm{Spec}(L)$, which has the structure of a $G$-torsor over $\mathrm{Spec}(K)$. At the level of fields, this ...
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Kummer-Dedekind's factorisation theorem

For a number field extension $K$ of $\mathbb{Q}$ one can factorise almost all prime ideals $(p)$ in the extension $K$, except finitely many, easily by factorising minimal polynomials in finite ...
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Prove that $a$ is a $p$th power in $k$ if only if it is in $K$

Let $k\subset K$ be an extension having degree $[K:k]=n$ coprime to $p$. Prove that $a$ is a $p$th power in $k$ if only if it is in $K$ This is a problem in Galois theory - Miles Ried. I'm learning ...
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Separable field extensions are Frobenius algebras

Wikipedia says that if $L/K$ is finite extension then $L/K$ is separable if and only if $L$ is a separable $K$-algebra. I am interested in the "only if" direction, which is outlined in the article. ...
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Sppliting field and Galois theory and its Automorphism group

I'm studying elementary Galois theory and came across these two questions: If $L = Gal(x^n-1, \mathbb{Q})$ then $Aut_{\mathbb{Q}} L$ is abelian. This question is followed by If $ L = ...
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Embedding of a field in a cyclic extension

Show that $K=\mathbb{Q}(\sqrt {a})$ for $a\in \mathbb{Z}$, $a<0$ can not be embedded in a cyclic extension whose degree over $\mathbb{Q}$ divisible by 4. I have tried for order exactly ...
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What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
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Norm of an element in cyclotomic extension (Exercises VI.19 Lang's Algebra)

Let $\zeta$ be a primitive $n^{\rm{th}}$ root of unity. Let $K=\mathbb{Q}(\zeta)$. If $n=p^r (r\geq 1)$ is a prime power, show that $N_{K/F}(1-\zeta)=p$ If $n$ is divisible by at least two distinct ...
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Equivalent conditions of a Galois extension (Exercise VI.4 in Lang's Algebra)

let $k$ be a field of characteristic $\neq 2$. Let $c\in k, c\notin k^2$. Let $F=k(\sqrt{c})$ . Let $\alpha=a+b\sqrt{c}$ with $a,b\in k$ not both $a,b=0$. Let $E=F(\sqrt{\alpha})$. Prove that the ...
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How is $Gal(\mathbb Q(\sqrt{2+\sqrt{2}})/\mathbb Q)\cong \mathbb Z/4\mathbb Z$?

$\def\Gal{\operatorname{Gal}}$ I was working on homework, and the problem starts off by saying that I previously showed (I can't find where, though) that with $\def\Q{{\mathbb Q}}\def\Z{{\mathbb ...
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The galois group of the polynomial $x^9+x^3+1$

What is the galois group of the polynomial $x^9+x^3+1$? Moreover give the bijection between subgroups and intermediate fields. Progress I think the order of the group is 108. But, there are many ...
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Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
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Prove that if $f\in R[X]$ , then $\displaystyle\prod _{\sigma \in G}f^{\sigma}\in \mathbb{Z}[X].$

Let $K$ be an algebraic number field and $R$ be the ring of algebraic integers of $K.$ Denote by $h^{\sigma}$ the polynomial obtained from $h\in K[X]$ after applying to its coefficients the ...
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Showing an algebraic element of $\bar{\mathbb{Q}}$ is in $\bar{\mathbb{Q}}$ [duplicate]

This question really has me stumped. We define $\bar{\mathbb{Q}}$ to be the set of elements in $\mathbb{C}$ that are algebraic over $\mathbb{Q}$. I have shown that this is a field. Now, I'm ...
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how much certain mathematician has been cited in recent articles

I want to know how much Galois work has been cited in recent articles. So I'm looking for a tool that can help me do that, more simply is there a tool to know how much certain mathematician has been ...
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Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
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Existence of irreducible polynomial over $\mathbb Q$, which has exactly two non-real roots

I know the following statement: If $f$ is an irreducible polynomial over $\mathbb Q$ of prime degree (let $\deg f=p$) and if $f$ has exactly two non-real roots in $\mathbb C$, then the Galois group ...
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Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
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Galois group of $x^{2^k}+1$

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?
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monomorphism on an algebraic field extension

let $E|K$ be an algebraic extension and $\phi:E\rightarrow E$ a $K $-algebra monomorphism,prove that $\phi$ is onto. i assume $\alpha\in E-\phi(E)$ to make a contradiction and i assume $f(x)$ to be ...
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Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
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product considering the period of index over a cyclotomic extension

This is an exercise from Milne Galois Theory (Chapter 3 Exercise 13). Let p be an odd prime, and let $\zeta$ be a primitive $p^{\text{th}}$ root of 1 in $\mathbb C$. Let $E = \mathbb Q[\zeta ]$, and ...
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Prove that this extension is Galois with galois group the quaternions

Let $M=\mathbb Q(\sqrt{2},\sqrt{3})$ and let $E=M(\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)})$. Prove that: M is Galois over $\mathbb{Q}$ Show that $E$ is Galois over $\mathbb Q$ with Galois group the ...
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galois group of a biquadratic involving primes.

Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral. I proved the ...
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Example of a non-separable normal extension

I'm trying to give an example of a normal field extension $K|F$ that is not separable. I now that if $F$ is finite or char$(F)=0$, $K|F$ is automatically separable, thus, I must look into infinite ...
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different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...
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248 views

Value in retracing mathematicians' steps (specifically Galois)?

So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus ...
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If $k[a_1,a_2,…,a_n]=k(a_1,a_2,…,a_n)$ show that $a_1,…,a_n$ are algebraic over $k$.

I am trying to prove the following statment and need some help. Let $k$ and $E$ be fields such that $k \subset E$ and $a_1,a_2, \ldots ,a_n \in E$, if $k[a_1,a_2,...,a_n]=k(a_1,a_2,...,a_n)$ show ...
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Cubic field extension question

Is the following statement true ? $$ 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}).$$
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Showing $\operatorname{Aut}(\mathbb{R})$ is abelian

I have an exercise (not for a class) that asks whether $\operatorname{Aut}(\mathbb{R})$ (field automorphisms) is abelian. I know that $\operatorname{Aut}(\mathbb{R})$ is just the trivial group, but ...
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Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
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Calculating a minimal polynomial over $\mathbb{Q}(\sqrt[3]{2})$

Let $L_1=\mathbb{Q}(\omega\sqrt[3]{2})$ where $\omega=e^\frac{2\pi i}{3}$ and $L_2=\mathbb{Q}(\sqrt[3]{2})$. I want to calculate $[L_1L_2:L_2]$, that it is the degree of the minimal polynomial over ...
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Calculate the degree of the field extensions.

I have been staring at this question for a while. I'm sure there is a little trick I am missing...anyway, it is the following: $ f = x^3 + x + 3 $ a) Show $f$ is irreducible over $\mathbb{Q}[x]$: ...
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Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
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Some Galois theory

I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ ...
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Irreducible factors of a polynomial in a Galois extension

Let $E|F$ be a finite Galois extension and $f(x) \in F[x]$ an irreducible polynomial. Prove that each of the irreducible factors of $f(x)$ in $E[x]$ have the same degree. An idea: Let $\phi \in ...
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Galois subextensions in a Galois extension

Let $F \subset E \subset L$ be fields such that $L/E$ and $E/F$ are both Galois extensions. Is $L/F$ necessarily a Galois extension?
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On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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Field extensions and irreducibility

I'm having trouble trying to show that the function f=x^3 + x + 3 is irreducible in the rationals. I tried using Eisensteins criterion but it didn't work as it doesnt satisfy all conditions. the ...
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Galois Extensions

I have to solve the following exercise: Let $f(x)=3x^4-3x^2+1$ be a polynomial in $\mathbb{Q}[x]$ and let $\alpha$ be one of its roots. 1) Determine $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and ...
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$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
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Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that ...
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Non-Galois number fields and complex embeddings

Let $K$ be a number field. $K$ is a normal extension of $\mathbb{Q}$ iff $\exists f(x)\in\mathbb{Q}[x]: K$ is the splitting field for $f(x)$. A field extension is Galois iff it is normal and ...
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Let $G = \text {Gal}(L/K)$. How do I see that $\phi \in G$ is completely determined by $\phi\mathbb( \sqrt[4] {3})$ and $\phi(i)$?

Let $K = \mathbb Q$ and $L = \mathbb( \sqrt[4] {3}, i)$. I see that $L \supset K$ is a an Galois extension. Also, after computations I've $[L : K] = 8$ (degree of $L$ over $K$ considered as a vector ...
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Deriving the Resolvent Cubic From Elementary Symmetric Functions

On page 614 of Dummit and Foote's Abstract Algebra, while deriving the resolvent cubic for a quartic $g$, they go from a trio of elements $\theta_i, i=1,2,3$ which were defined in terms of the roots ...
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relation between the characteristic polynomial and the minimal polynomial

Define $l_a : F(a) → F(a) $ by $ l_a(x)=ax$, when $[F(a):F]=n$ . show that the minimal polynomial of $a$ over $F$ is the same as the minimum polynomial of $l_a$ as defined in linear algebra. this ...
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a question about field extensions and tower formula

if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction ...
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On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
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an exercise in Galois theory about polynomials

find a field $F$ and different polynomials $f(X),g(X)\in F[X]$ that for every $\alpha \in F$ we have $f(\alpha)=g(\alpha)$. prove that it is impossible if $F$ is infinite. i think this example ...