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Questions tagged [galois-theory]

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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the cyclic group Gal(R(x)/R) has order 3 where R(x) is the field of quotients of R[x]

lets consider the field $\mathbb{R}(x)$ formed by the quotients of $\mathbb{R}[x]$. we know that $A=\begin{pmatrix} \frac{-1}{2} &\frac{-\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{-1}{2} \end{...
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How to check if permutation induces an element of the Galois group.

Let $f \in \mathbb Q[X]$ be irreducible of degree $n$ with zeros $\alpha_1,\dots,\alpha_n \in \mathbb C$. Further, let $L$ be the splitting field of $f$ and $\sigma \in S_n$. Is there an easy way to ...
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51 views

Subfields of $\mathbb{C}$ Galois extension

Let E be a subfield of $\mathbb{C}$, which is a Galois-extension of $\mathbb{Q}$. Let $E_0=E \cap \mathbb{R}$. Show that: i) $[E:E_0]\leq2$ ii) Is $[E_0:\mathbb{Q}]$ always a Galois-extension? iii) ...
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Rank deficient LDPC code

I am new to working on LDPC codes and have gotten stuck on this issue for some time. I am struggling on how to deal with parity check matrices that are not of full rank. I generate the parity check ...
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2answers
49 views

Degree of splitting field of $x^5-1$ over $\mathbb{Q}$ and the order of its Galois group?

If I understand the result here correctly, the order of the Galois group is equal to the degree of the extension. That said, I am fairly certain that the splitting field $E$ of $x^5-1$ over $\mathbb{Q}...
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19 views

Field Extension $k(t^{1/n})/k(t)$ Cyclic

Let $k$ be a field. We consider the field extension $k(t^{1/n})/k(t)$. My question is why and how to see that it is cyclic, therefore $Gal(k(t^{1/n})/k(t))$ is a cyclic group. There is one case that ...
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1answer
59 views

Integral Closure, Galois extension,and Dedekind Domain

Let $A$: Dedekind domain, $K$: $\operatorname{Frac}(A)$, $B$: Dedekind domain with $A \subset B$, $L$: $\operatorname{Frac}(B)$ Let $L/K$: galois extension with galois group: $G$. $B^G=\{b \in B \...
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2answers
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Order of $\operatorname{Gal}(K_s/K_\ell)$

I am reading the proof of Grothendieck’s proposition about $\ell$-adic representations of the decomposition group of some discretely valued field, the proposition in the appendix of Serre and Tate’s ...
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1answer
19 views

What automorphism cannot be extended when the extension is not normal?

As I understand it, if we have an automorphism $\phi : K \rightarrow K$, and a finite normal extension $N/K$, $\phi$ can always be extended to an automorphism of $N$. But what happens if $N/K$ is not ...
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Can a degree $p^n$ field extension always be factored in a sequence of prime extensions?

Suppose $L/K$ is a field extension of degree $p^n$ for some prime $p$ (if necessary, assume the characteristic of $K$ is not $p$). Then, is it always possible to find a sequence of extensions $K = ...
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25 views

Showing a subfield is fixed under complex conjugation

Suppose we let $\sigma: \mathbb{C} \to \mathbb{C}$ be the complex conjugate map, and we have a subfield $L \subseteq \mathbb{C}$ with $L/\mathbb{Q}$ a finite Galois extension. I want to show that $\...
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Finding a basis for a field extension $\mathbb{Q}(i,\sqrt[4]{2})/\mathbb{Q}(\sqrt{2}).$

I am looking to find a basis for the field extension $$\mathbb{Q}(i,\sqrt[4]{2})/\mathbb{Q}(\sqrt{2}).$$ Clearly as far as I can tell the elements $i$ and $\sqrt[4]{2}$ would be in the basis, as ...
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41 views

factors of $X^{p^n}-X$

I'm my study of Galois theory I have been struggling with the following proposition without much success: The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible ...
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1answer
48 views

$\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic [duplicate]

How to prove that $\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic. I thought that they are but I got this problem in Dummit Foote Section 14.1. Question no 4. As they extension over $\Bbb Q$...
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1answer
51 views

Describing the Galois group of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ without computing the extension degree (proof check please)

An exercise asks to describe (i.e. basically tell what it is isomorphic to, rather than listing the automorphisms explicitly) the Galois group of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ and suggests computing ...
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1answer
57 views

Galois group of $X^5-5$

I am trying to calculate this galois group of $x^5-5$ over $\mathbb Q$. I know that there is a tower of extensions with groups $C_4$ and $C_5$ respectively, so the group is order 20. I am guessing it ...
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1answer
64 views

Let $K = F_2(x)$ and $q_t(x) = x^2-x-t \in K[x]$. Show that $q_t$ is not solvable by radicals.

Let $K = F_2(x)$ and $q_t(x) = x^2-x-t \in K[x]$. Let $E$ be its splitting field. I need to prove that $Gal_{E/K} \simeq \mathbb{Z_2}$ and that $q_t$ is not solvable by radicals. Any hint?
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1answer
60 views

decomposition group and inertia group, the minimal polynomial,surjectivity of the map $D_{M/P}\rightarrow Gal$

Can anyone explain the underlined sentence? For notation, A:Dedekind domain, K=Frac(A), L/K:Galois extension, B:The integral closure of A in L, M:A maximal ideal of B, P:The intersection of M and A (...
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2answers
86 views

Inverse Galois Problem for Direct Product of Groups

Assume that $G = \operatorname{Gal}(L/\Bbb{Q}), H = \operatorname{Gal}(K/\Bbb{Q})$. Can we construct a field extension of $\Bbb Q$ from $L$ and $K$ such that $G \times H$ is its Galois group? For $...
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1answer
44 views

Splitting fields for $x^3-3$ and $x^5-1$

I'm looking for the splitting fields of (a) $x^3-3$ (b) $x^5-1$. EDIT: (a) Thanks to all the hints and suggestions, the three roots are $x_1=3^{\frac{1}{3}}$, $x_2=e^{\frac{2 \pi i}{3}}3^{\frac{...
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30 views

Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book and it says that for $A = $ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
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1answer
39 views

Let $K/\mathbb Q$ be a number field and a Galois extension. Show $K$ contains the field $\mathbb Q(\sqrt{d_K})$ where $d_K = \mathrm{disc}(K)$.

Let $K/\mathbb Q$ be a number field and a Galois extension. Show $K$ contains the field $\mathbb Q(\sqrt{d_K})$ where $d_K = \mathrm{disc}(K)$. I'm guessing this will somehow use the fact that the $...
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3answers
42 views

Does there exist a prime $p$ such that $X^4+X+1$ splits into a product of irreducible quadratics over $\mathbb F_p$?

Does there exist a prime $p$ such that $X^4+X+1$ splits into a product of irreducible quadratics over $\mathbb F_p$? I have checked a few primes but I just get a single linear factor out, or it is ...
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2answers
60 views

Compute the irreducible polynomials over Q for $a=\sqrt{2}+\sqrt{5}$ and $b=\sqrt[3]{2}+\sqrt{5}$

Compute the irreducible polynomials over Q for $a=\sqrt{2}+\sqrt{5}$ and $b=\sqrt[3]{2}+\sqrt{5}$ For a, I do: $$a=\sqrt{2}+\sqrt{5} \Rightarrow a^2-7=2\sqrt{10} \Rightarrow a^4-14a^2+9=0$$ So $p(X)...
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When is the restriction map $Gal(L/F) \rightarrow Gal(K/F)$ a surjection?

I was wondering, when the restriction map $Gal(L/F) \rightarrow Gal(K/F)$ is a surjection? I found a good answer to this question here. In point 3), Starfall explains why with the hypothesis of the ...
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$F$-Automorphism of $F(w)$ induces automorphism on $\{1,w,w^2,…,w^{n-1} \}$ only if $w$ primitive n-th root?

I just stumble across a theorem in Galois theory, that says that if F is a field of characteristic p and $w$ is a primitive n-th root of unity with $p \nmid n$ than if we take a $F$-automorphism $\...
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2answers
56 views

Galois group of $f(x)=x^5+2x+1\in\mathbb{Z}_3[x]$

consider $f(x)=x^5+2x+1\in\mathbb{Z}_3[x]$,what is the splitting field of $f$ and its Galois group? I know it is a Galois extension. But i do not know the degree of the extension for i have no idea ...
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1answer
20 views

A particular field generated from a set

I am trying to understand a problem from my book regarding field extensions and fields generated from sets. I have shown the set $B_0 = \lbrace (0,0),(1,0) \rbrace$ to generate a field $Q$ which ...
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Question about Galois group of a splitting field [duplicate]

I have a question about Galois group. Actually, we are considering : $P(X) = X^6-5$. The roots on a algebraic closure are : $X = \{5^{\frac{1}{6}}, z_65^{\frac{1}{6}}, ..., z_{6}^55^{\frac{1}{6}} \}$, ...
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1answer
38 views

Mistake (?) in differential Galois theory

I have found this exercise in the book of Crespo and Hajto “Algebraic groups and differential Galois theory”: let $\mathcal{L}(Y):=Y^{(n)}+a_{n-1}Y^{(n-1)}+\dots+a_1Y’+a_0=0$ and let $W$ denote the ...
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1answer
47 views

Why isn't the the galois group of a polynomial with n distinct roots isomorphic to Sn?

If we consider the polynomial $f(x)=x^3-3x-1 \in \mathbb{Q}[x]$ which has 3 real roots $\{x_1,x_2,x_3\}$. I read that its galois group is isomorphic to $A_3$ and not to $S_3$. I don't really ...
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1answer
40 views

Proving a subgroup of a Galois group is normal

"Let $N/K$ be a finite Galois extension with Galois group $G = Gal(N/K)$. Let $M$ be an intermediate field of the extension, let $E = Gal(N/M)$, and let $F = \cap_{\sigma \in G} σEσ^{-1}$ . Show ...
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1answer
34 views

Galois Group of a Quadratic polynomial

"Let N be the splitting field of $f(X)=X^4+9$ over $\mathbb{Q}$, and let $G = Gal(N/\mathbb{Q})$. Using the fact that $α = \sqrt{3}\exp{\frac{πi}{4}}$ is a root of $f(X)$ and $α^{2} = 3i$, show that $...
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47 views

Fields such that all finite extensions are Galois

What can we say about a field $F$ such that any finite extension $K/F$ is Galois? Clearly, $F$ is perfect. For instance, it seems to hold if $F$ is quasi-finite or $[\overline F : F] < \infty$. ...
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1answer
60 views

Abelian Galois group of even order

I'm stuck at the following problem. Let $a$ and $b$ be algebraic real numbers over $\mathbb{Q}$. Let $K= \mathbb{Q}(a+bi)$ be a simple extension of $\mathbb{Q}$. Suppose that $K$ is a Galois ...
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51 views

Solubility of general cubic by radicals

My task is to use the following two theorems to prove that any cubic is soluble by radicals. We are not allowed to use that $\Bbb S_3$ is a soluble group. Theorem 1: Hilbert's Theorem 90 Let $L:K$...
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3answers
180 views

How to prove that $(-18+\sqrt{325})^{\frac{1}{3}}+(-18-\sqrt{325})^{\frac{1}{3}} = 3$

How to prove that $\left(-18+\sqrt{325}\right)^{\frac{1}{3}}+\left(-18-\sqrt{325}\right)^{\frac{1}{3}} = 3$ in a direct way ? I have found one indirect way to do so: Define $t=\left(-18+\sqrt{325}\...
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0answers
38 views

Determine $[K:\mathbb{Q}]$ and show that $Gal(K/\mathbb{Q})$ is non-abelian.

Question: Let $K$ be a splitting field for $x^5-2$ over $\mathbb{Q}.$ $(1)$ Determine $[K:\mathbb{Q}].$ $(2)$ Show that $Gal(K/\mathbb{Q})$ is non-abelian. For $(1),$ note that roots of $x^...
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1answer
38 views

How to prove $ \left\{ t^2,t^3 \right\}$ equals the vanishing set of $y^2-x^3$?

Exercise 3.2 in Hartshorne is about proving that morphisms of varieties may be underlain by homeomorphisms without being isomorphisms of varieties. The morphism in consideration is $\varphi:t\mapsto (...
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2answers
61 views

Construction of cyclic local field extensions of arbitrary degree and ramification index

Let $K$ be a local field. Let $n$ be an arbitrary natural number and $e$ be any divisor of $n$. Question Does there exist an extension $L/K$ with the following properties? $L/K$ is a cyclic ...
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0answers
39 views

Proof of the main theorem on non-abelian Kummer extensions (following Lang)

I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions: In the proof of 11.1, Lang says that the ...
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1answer
76 views

Trisecting $2\pi/5$, is this possible?

I guess that the answer is no, even knowing that $cos(2\pi/5)$ is constructible since the $5$th root o unity is construtctible. But when I use the trick for finding the minimal polynomial of $3\theta=...
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1answer
17 views

Exact Sequence of Galois Groups

Let $E_1/F$, $E_2/F$ be Galois extensions. Then $E_1E_2/F$ and $E_1\cap E_2/F$ are Galois extensions. Supposedly there is a short exact sequence $$1\to \mathrm{Gal}(E_1E_2/F) \xrightarrow{\varphi} \...
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1answer
63 views

Galois extension - minimum polynomial

Let $K$ be a Galois extension of $F$ and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=\text{Gal}(K/F)$ and $H=\text{Gal}(K/F(a))$. We symbolize with $\tau_1, \ldots , \tau_r$ the representatives of ...
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0answers
55 views

There is an intermediate extension of degree $p$

Let $K$ be a Galois extension of $F$ and $p$ be a prime factor of the degree $[K:F]$. I want to show that there is an intermediate extension $F\subseteq L\subseteq K$ with $[K:L]=p$. $$$$ Do we ...
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2answers
66 views

Constructing a polynomial given the Galois Group of it's splitting field

Let $red_p : \mathbb{Z}[x]\to\mathbb{Z}/(p)[x]$ be the canonical ring morphism sending a polynomial with integer coefficients to a polynomial with integer coefficients modulo $p$, with $p$ a prime, by ...
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1answer
27 views

Existence of polynomials in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$ with same splitting fields.

I've been asked to prove the following statement in a Galois Theory Seminar after being introduced to Dedekind's Theorem. (I assume this could potentially help getting the answer.) Let $f(x)$ be a ...
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1answer
42 views

Show that $\mathbb{Q}(\alpha+i\alpha)/\mathbb{Q}$ is not Galois

Question: Let $\alpha$ be a real root of $x^4-5 \in \mathbb{Q}[X].$ Show that $\mathbb{Q}(\alpha+i\alpha)/\mathbb{Q}$ is not Galois extension. To show that an extension is not Galois, we just need ...
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1answer
63 views

Determine the automorphism group $Aut(\mathbb{Q}(\sqrt{13}, \sqrt[3]{7})/\mathbb{Q})$

Question: Determine the automorphism group $$Aut(\mathbb{Q}(\sqrt{13}, \sqrt[3]{7})/\mathbb{Q}).$$ My attempt: Since the polynomial $(x^2-13)(x^3-7)$ has roots $$\sqrt{13}, -\sqrt{13}, \sqrt[3]{7}, \...
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0answers
41 views

Find polynomial given splitting field

Let $f\in\mathbb{Q}[x]$ a monic polynomial such that $f$ has degree $n$. Let $E_f$ be the splitting field of $f$ over $\mathbb{Q}$. I would like to show that there exists a monic polynomial in $\...