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Questions tagged [galois-extensions]

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For questions about Galois extensions fields. We say $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

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Computing a certain Galois group

I am trying to answer this question: Suppose $p$ is an odd prime and $K/\mathbb{Q}$ is the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of 1 in $\mathbb{C}$. (a) Show that $K$...
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How can you specifically extend an automorphism from a quadratic field to one of a cyclotomic field?

I am trying to see how can I extend the automorphisms of the Galois extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$, for $d$ square-free, to automorphisms of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ that fix $\...
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37 views

$L|_k$ be a finite Galois extension and $M,N$ are subfields containing $k$ such that $[M:k], [N:k]$ are powers of $2$. Is $[MN:k]$ power of $2$?

Let $L/k$ be a finite Galois extension, and let $M$ and $N$ be subfields containing $k$ such that $[M:k]$ and $[N:k]$ are powers of $2$. Is $[MN:k]$ a power of $2$? I think this is false, but I have ...
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Galois group of $x^5-5x^3+4x+1 \in \mathbb{Q}[x] $

Can the Galois group of $x^5-5x^3+4x+1 \in \mathbb{Q}[x] $ be isomorphic to $S_5$? I know that it has five real roots, then I think that it is impossible. I think that this Galois group cannot contain ...
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is $f(x) = 20x^5-35x^4+10x^3-1$ irreducible over $\Bbb Q [\sqrt{3}]$?

I was studying for a Galois theory exam when I saw this question and was stumped. It is not hard to check that $f$ is irreducible over $\Bbb Q$ (since $x^5\cdot f(\frac{1}{x})$ is irreducible over $\...
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If $f(t,x)\in F_q(t)[x]$ is a Morse function, does this mean splitting field of $f(t,x)$ over $F_q(t)$ is a regular extension?

One of the classical result of Hilbert says, if $f$ is a Morse function, then the splitting field of $f(X,T)$ over $Q(T)$ is a regular extension with Galois group $S_n.$ J. P.Serre- Topics in Galois ...
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58 views

Why doesn't “$S_n$ appears as a Galois group over $\Bbb{Q}$” wrap up the Inverse Galois Problem?

Since every finite group $G$ is embedded in $S_n$ for $n = |G|$ and Hilbert showed that $S_n$ appears as a Galois group of $K/\Bbb{Q}$ for some Galois extension $K$, then how does that not wrap up the ...
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$\mathbb{Q}(\alpha)$ extension of degree 3 is galois over $\mathbb{Q}$ if and only if discriminant of minimal polynomial of $\alpha$ is square.

I supposed that $\alpha$, $\beta$ and $\gamma$ are the roots of the minimal polynomial in its splitting field. So the discriminant is $(\alpha-\beta)^2(\alpha-\gamma)^2(\beta-\gamma)^2$. If every ...
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Hopf-Galois structures of cyclic type on a dihedral or quaternionic extension

Let $L/K$ be a dihedral or quaternionic finite field extension, that is such that $Gal(L/K)$ is either a dihedral or a quaternion group. How many Hopf-Galois structures of cyclic type are there on ...
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1answer
63 views

$K$ be a finite extension of $\mathbb Q$ then there exists a finite extension $E$ over $K$ such that $E/K$ is not normal.

Let $K$ be a finite extension of $\mathbb Q$. Then I have to show that there exists a finite extension $E$ over $K$ such that $E/K$ is not normal. It is clear that $E/\mathbb Q$ is not normal as well....
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51 views

Adding finite list of square roots of primes to $\mathbb{Q}$

I have that idea which I'm pretty sure that is true- but I haven't succeded to prove it: Given finite list of primes ${p_1,p_2,p_3,...,p_n}$ , and extension of the rational field with the square roots ...
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3answers
111 views

Finite Galois group of Galois extension implies that the extension is finite?

Assume that the field extension $K \subset L$ is a Galois (in other words: normal and separable, possibly infinite) extension with finite Galois group. If one starts with a finite Galois extension, ...
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1answer
35 views

Is the extension of a perfect field by one of its algebraic closures always a Galois extension?

Let $\overline{K}$ be an algebraic closure of a perfect field $K$. I have read that the absolute Galois group of a perfect field is simply $\text{Gal}(\overline{K}/K)$. Yet if Galois groups are only ...
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79 views

Is $\mathbb{Q}(\cos\frac{2\pi}{13})/\mathbb{Q}$ a Galois extension?

I've been solving problems from my Galois Theory course, and I got stuck in this problem: Calculate the number of subfields of $\mathbb Q(\cos\frac{2\pi}{13})$ What I considered doing is finding the ...
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1answer
58 views

Showing the extension $\mathbb{Q}(\sqrt[6]{-3})/\mathbb{Q}$ is Galois and determining its Galois group.

I'm trying to show that $\mathbb{Q}(\sqrt[6]{-3})/\mathbb{Q}$ is a Galois extension. I would like to show that $\mathbb{Q}(\sqrt[6]{-3})/\mathbb{Q}$ is the splitting field of $f(x) = x^6 - 3$ which is ...
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1answer
71 views

What is $\text{Frob}\in \text{Gal}(\overline{K}/K)$?

Let $\overline{\rho}: \text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{F}_q)$ be an unramified Galois representation at a place $v$ of $K$. Since $\ker(\overline{\rho})$ is closed, it corresponds to ...
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1answer
27 views

How to ensure the generator of Galois extenion?

I meet an exercise when I study Galois Theory: Suppose $E/F$ is a finite Galois extension. Suppose $a\in E$ $\\$ is an element fixed only by the identity automorphism of E, i.e.$\{\phi\in \rm{Aut}(E):\...
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Find $Gal(\mathbb{Q}(e^{2\pi i/6},e^{2\pi i / 10}) / \mathbb{Q})$

First I need to show that the extension $\mathbb{Q}(e^{2\pi i/6},e^{2\pi i / 10}) / \mathbb{Q}$ is actually Galois extension, and then I need to find the group itself. Denote $\alpha = e^{2\pi i/6} $, ...
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How to prove whether $i\in\mathbb{Q}(\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5})$ or not

I've been solving problems from my Galois Theory course, and I need help with some detail in this one. It says: Calculate how many subfields has the splitting field of $P=X^7+4X^5-X^2-4$ over $\...
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1answer
40 views

Prove $\operatorname{Gal}(M_3/M_1)\cong\operatorname{Gal}(M_2/(M_1\cap M_2))$

I've been solving problems from my Galois Theory course, and I got the idea to solve this one but I'm unable to prove some details. The problem goes like this: Being $L/K$ a Galois extension, and ...
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Normal closure as the compositum of conjugates

An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242. Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
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1answer
46 views

Theorem 17.22 from Ian Stewart's Galois Theory

Apply the Frobenius map to minimal polynomials to see that $$ [K(\alpha^p+\beta^p):K(\alpha^p+\beta^p,\beta^p)]\leq [K(\alpha+\beta):K(\alpha+\beta,\beta)] $$ and $$ [K(\alpha^p+\beta^p):K] \leq [K(\...
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1answer
45 views

Showing $\mathbb{Q}(\sqrt 2, \sqrt 3 ) = \mathbb{Q}(\sqrt 2 + \sqrt 3)$ using information about Galois group

Let $E/F = \mathbb{Q}(\sqrt 2, \sqrt 3 )/\mathbb{Q}$. This extension is Galois of degree 4, and one can easily verify that $G:= \operatorname{Gal}(\mathbb{Q}(\sqrt 2, \sqrt 3 ) /\mathbb{Q})$ is $\{ \...
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146 views

Is this a valid proof for the normalness of subgroups in galois theory?

Let $F/\mathbb{Q}$ be a finite galois extension, and $Aut(F/\mathbb{Q})$ be the corresponding galois group. What I want to show is that for all intermediate fields between $F$ and $\mathbb{Q}$, where $...
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49 views

Degree of a polynomial zero over the simple extension attained by adjoining another zero.

The original question that I set out to answer is as follows: Let $\mathbb{Q}$ be the field of rational numbers, let $p \in \mathbb{Q}[x]$ be a monic, irreducible polynomial with $n$ distinct roots $...
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38 views

Splitting field of a degree 4 irreducible polynomial in $\mathbb{Q}[x]$ with two real roots and two non real roots

Let $f \in \mathbb Q[x]$ be an irreducible polynomial of degree $4$, such that two of its roots are in $\mathbb R$ and two are in $\mathbb C \setminus \mathbb R$. Let $E \subseteq \mathbb C$ be the ...
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1answer
39 views

splitting field of separable and irreducible polynomial: does isomorphism of Galois subgroups imply isomorphism of subfields?

The original problem is here: Galois extension: does isomorphism in subgroups implies isomorphism of the subfield?. My question was, given $L/Q$ finite Galois extension, and suppose for $S$, $H$ ...
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Equivalent Definitions of Field Trace of Galois Extension

When studying trace and norm of finite Galois extensions, I stumbled upon two (for Galois extensions) equivalent definitions. $Tr_{L|K}(a) = tr(\phi_a)$ with $\phi_a: L \to L, x \mapsto ax$ $Tr_{L|K}(...
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1answer
65 views

E/F is Galois ext with $[E:F]=p^2$ and its Galois group is not cyclic, then $\exists K$ a proper subfield of $E$.

Let $E$ be a Galois extension of a field $F$. Suppose that the Galois group $Gal(E/F)$ is an abelian group of order $p^2$ which is not cyclic. Then I want to show $\exists K$ a proper subfield of $E$...
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43 views

Proving the irreducibility of a polynomial based on its Galois group

Suppose $f(X) \in \mathbb{Q}[X]$ is a polynomial of degree $n$. Let $K$ be the splitting field of $f(X)$ over $\mathbb{Q}$. Prove that if $Gal(K/\mathbb{Q}) \simeq S_n $, then $f(X)$ is irreducible ...
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1answer
38 views

What does it mean when one talks about splitting field of a multivariable polynomial? And then, Galois group of that splitting field?

I came across the following, while I was reading a recent research article. I do not know how to interpret it; the article does not define this concept (perhaps because it is too elementary). Could ...
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1answer
46 views

Criterion for finite Galois extension

I am trying to relax the condition for checking whether a extension is Galois. I got from a textbook that: $E/F$ is a finite Galois Extension iff $E$ is a spitting field of a separable irreducible ...
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4answers
200 views

How can I prove that $\sqrt{2}$ is not inside Q($\sqrt{5 + \sqrt{5}}$)?

I want to prove it, but although it is quite intuitive I don't know how to prove it mathematically. I have tried using the tower rule but I haven't got anything crear. Thanks in advance.
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A question related to abelian extensions

Let $K/F$ be an abelian extension (a Galois extension $K/F$ such that $Gal(K/F)$ is abelian) of degree at least $2$. Prove that there exists a tower of fields $F=K_{0} \subset K_1 \subset \cdots \...
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Galois group for $\operatorname{Gal}(\mathbb{Q}(\zeta_n + \zeta_n^{-1})/\mathbb{Q})$

I want to find, classify $\operatorname{Gal}(\mathbb{Q}(\zeta_n + \zeta_n^{-1})/\mathbb{Q})$ group up to isomorphism where $\zeta_n = e^{\frac{2\pi i}{n}}$. I know $\zeta_n$ is a root of $x^2 - (\...
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1answer
95 views

Existence of a Field not containing $\sqrt{2}$, such that any finite extension is cyclic.

I want to show the existence of a field $E$ not containing $\sqrt{2}$, such that any finite extension of $E$ in $\overline{\mathbb{Q}}$ is cyclic. I think the maximal field not containing $\sqrt{2}$ ...
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1answer
47 views

Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$.

I found the following problem: Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$. Can someone give me a hint? I remember ...
3
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1answer
76 views

Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?

I am studying these notes and I am trying to generalize a bit the setting of the Section 3, because there doesn’t seem to be a fundamental reason to only study $p$-adic fields. So all the fields ...
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0answers
29 views

Still confused on splitting fields and Galois extension

Let $F = \mathbb{Q}$ and $E = \mathbb{Q}(\sqrt[4]{2})$. Then, $E$ is an extension of $F$. Since if $E$ is the splitting field of $f(x)$ over $F$, then $E$ is galois extension of $F$. But why $E$ is ...
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45 views

Elements of Galois group map roots of minimal polynomial to which other roots?

If we have $F \leq E \leq \mathbb{C}$, I know that for any $\alpha \in E$ and $\sigma \in \operatorname{Gal}(E/F)$, we have that $\sigma$ maps roots of $m_{\alpha,F}(x)$ to roots of $m_{\alpha,F}(x)$. ...
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1answer
54 views

Existence of an irreducible quartic polynomial in $\mathbb{Q}[x]$ with four real roots and Galois group $A_4$.

There is an example of an irreducible quartic with rational coefficients whose roots are all real and whose Galois group is $S_4$. Is there a similar example of an irreducible quartic $f$ in $\mathbb ...
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38 views

Is there an irreducible quartic over $\mathbb{Q}$ whose splitting field is not radical?

It is well known that there are degree 3 irreducible polynomials over $\mathbb Q$ whose splitting fields are not radical. Indeed, there are examples where the splitting field is a real Galois ...
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85 views

Prove that if $\alpha \in E$ and $f(\alpha) = 0$, then $E = E^H(\alpha)$.

Let $F \leq C$. Suppose that $f(x) \in F[x]$ is monic, irreducible over $F$ and $\deg f(x) = 6$. Let $E$ be the splitting field of $f(x)$ over $F$ and let $G = \text{Gal}(E/F)$. Assume that $[E:F]=12$ ...
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72 views

Problem with this Galois Group

Consider $$K:=\Bbb{Q}(\{\root n\of2\mid n≥2\})⊆\Bbb{R},$$ find $Gal(K/\Bbb{Q})$. So it would be at most $n!$ (infinite) number of $\Bbb{Q}$- automorphism over K. I should apply the simple extension ...
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1answer
87 views

Use Galois theory to prove that $H \triangleleft G$

Suppose that $F \leq E \leq \mathbb{C}$, $[E : F] = 100$, $E$ is Galois over $F$ and $G = Gal(E/F)$ contains a subgroup $H$ such that $|H| = 25$. Use Galois theory to prove that $H \triangleleft G$. $\...
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40 views

Proof that if you have two splitting fields for the same polynomial and one of them is contained in a radical extension, then so is the other one.

Let $E|K$ and $E'|K$ be splitting fields of a polynomial $f \in K[X]$. Prove that if $E$ is contained in a radical extension of $K$, then so is $E'$. My "proof": $E \cong K(R(f))$, where $R(...
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1answer
66 views

Textbook's proof of Galois correspondence is circular

I am reading Galois Theory 4th edition by Ian Stewart and the proof of one part of the Galois correspondence is circular. $L : K$ is a finite normal field extension inside $\mathbb{C}$ with Galois ...
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1answer
75 views

Question regarding galois extension and galois groups

From what I have read, for a field extension $E/F$, the automorphism group $Aut(E/F)$ is a Galois group when $E/F$ is Galois, that is, the extension is normal and separable, or equivalently, when $E$ ...
3
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0answers
54 views

Field Extensions with Common Transcendental Numbers

Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For ...
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1answer
48 views

Is the quotient field of $\mathbb{Z}[\sqrt{-3}]$ galois over $\mathbb{Q}$ and what is the Galois group?

I proved that $\mathbb{Z}[\sqrt{-3}]$ is isomorphic to $\mathbb{Z}[x]/(x^{2}+3)$. Because $f = (x^{2}+3)$ is irreducible by Eisenstein. Is that enough to conclude that it splits over $\mathbb{Q}$? If ...

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