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-connections).

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Checking if the intersection of two cyclic $p$-adic extensions with certain properties is trivial

Let $L$ and $L'$ be finite extensions of $K = \mathbb{Q}_p$. Also, let $n = [L:K]$ and $e = e(L/K)$. Furthermore, we assume the following properties: $L$ and $L'$ are both cyclic over $K$, $L'/K$ is ...
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Infinite sums for quintic polynomial

It is well known that there is no finite solution for the roots of a quintic polynomial. Are there any nice formulas in terms of infinite sums? Clearly the definition of nice is important for the ...
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(Galois)group of order max$\{m,n\}$

Let $K$ be a field, $p, q ∈ K[X]$ coprime with deg $p = m$, deg $q = n$, and suppose $f := \frac{p}{q} ∈ K(X)\setminus K$. Supposedly, then, $K(X)/K(f)$ is an algebraic extension of degree max$\{m,n\...
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The classical Gauss sum

This is a problem from dummit and foote 14.7.11 I solved everything except b). I tried hard but couldn't solve it. I'm trying to using sum of pth root of unity is 0. But there are $p$ terms in ...
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Splitting of primes and other properties of $\mathbb{Q}[\omega]$ for $\omega=e^{2\pi i/m}$

Reading through Marcus I came to this exercise part of which already have answers in this same site (Splitting of primes in real cyclotomic field ) but no complete answer can be found and I'm having ...
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Restriction of the Frobenius automorphism for normal extensions

I'm studying number theory on Marcus book and at a certain point I'm required to prove the following facts about the Frobenius automorphism. We start with a lemma and then are required to specialize ...
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Find Galois group of $\mathbb{Q}\sqrt{(2+\sqrt2)(3+\sqrt3)}$ [duplicate]

Find Galois group of $Q=\mathbb{Q}\sqrt{(2+\sqrt2)(3+\sqrt3)}$. I know the minimal polynomial is $f(x)=x^8 -24x^6+144x^4-288x^2+144$. It's irreducible over $\mathbb{Q}$, hence, $[Q:\mathbb Q]=8$. ...
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53 views

How is this function injective?

I'm currently studying Galois Theory and I came across this theorem. Theorem Let $E$ be a field, $p(x)\in E[x]$ an irreducible polynomial of degree $d$ and $I = \langle p(x) \rangle$ the ideal ...
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1answer
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Elements of the subgroup for Galois group over $\mathbb{Q}$ (Cyclotomic extension)

Well... this question looks like silly though, I have a curious about the below. Let $K= \mathbb{Q}(\omega)$ for $\omega=e^{2\pi i \over n}$ My book said $G(K/\mathbb{Q}) = \{\sigma_i \vert \...
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How does Discriminant of splitting field of irreducible polynomial $f$ related to discriminant of $f $ ??

Can we express Discriminant of splitting field of polynomial $f$ in terms of the discriminant of $f .$
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Finding a fixed polynomial under the multiplicative inversion automorphism

Can anyone find a polynomial $f ∈ ℚ\left(X+\frac{1}{1-X} + \frac{X-1}{X}\right) ⊆ ℚ(X)$ that is fixed under the automorphism $(X ↦ \frac{1}{X})$? $f = X+\frac{1}{X}$ would be nice, but I don't know ...
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Show that $H_1$ and $H_2$ are conjugate subgroups of $G=\text{Gal}(L/K)$.

Suppose that $L$ is a finite Galois extension of $K$, $f$ is a monic irreducible polynomial in $K[X]$, and $\alpha_1$ and $\alpha_2$ are elements of $L$ such that $f(\alpha_1)=f(\alpha_2)=0$. Prove ...
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39 views

What finite groups can appear as collections of automorphisms of some field? Proof verification

What finite groups G can appear as collections of automorphisms of some field? More precisely, for which G does there exist a field F such that G is a subgroup of the automorphism group of F? What ...
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1answer
68 views

An extension corresponding to a subgroup of Galois group

Let $G$ be the Galois group of $f(x)=x^6-2x^4+2x^2-2$ over $\mathbb{Q}$. Describe an extension corresponding to any of it's proper subgroups of maximal order (i.e. find generators of this extension). ...
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Relation between subgroup topology and Krull topology for an intermediate field of a Galois extension

Let $E/K$ be a Galois extension and let $F$ be an intermediate field such that $K\subseteq F\subseteq E$. Then $E/F$ is a Galois extension too and $H=\mbox{Gal}(E/F)$ is a closed subgroup of $G=\mbox{...
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$K(\zeta)/k(\zeta)$ is Galois (proof verification)

Let $K/k$ be Galois and $\zeta$ be primitive $n$-th root of unity. I want to prove that $K(\zeta)/k(\zeta)$ is also Galois. Proof: Recall that an extension is Galois iff it is a splitting field of ...
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32 views

Find the primitive element of extension

Find the primitive element of extension $Q<Q(\sqrt{2}-i, \sqrt{3}+i) $ I am asking for help how to find a primitive element, please
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Using symmetric polynomials to find the discriminant of $x^4 + px + q$ over $\mathbb{Q}$

I'm trying to prove that the discriminant of $x^4 + px + q$ over $\mathbb{Q}$ is $-27p^4 + 256q^3$, where we define the discriminant to be $$ \Delta_f = \prod_{i < j}(\alpha_i - \alpha_j)^2 $$ I ...
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61 views

How to find the degree of the extension $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$?

How to find the degree of extension for $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$? I believe that the minimal polynomial of $\sqrt[4]{3+2\sqrt{5}}$ is $x^8-6x^4-11$, but I don't know how to ...
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1answer
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If $\text{Gal}(K/\mathbb{Q})\cong Z_5$, then show $K(\sqrt{2})/\mathbb{Q}$ is Galois

a) Find a field extension $K/\mathbb{Q}$ such that $\text{Gal}(K/\mathbb{Q})\cong Z_5$ ($Z_5$ denotes the cyclic group on $5$ elements). b) Let $L=K(\sqrt{2})$. Show $L/\mathbb{Q}$ is Galois and ...
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Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$

Find Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$ and describe an extension corresponding to any of it's proper subgroups of maximal order. I know that the roots are $$\sqrt{\frac{1}{3}\left(2 ...
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Examples of quadratic extensions K, L of $\mathbb{Q}$ such that KL has some properties.

Let $p$ be a prime integer, I want to find $p$ and K, L extensions of $\mathbb{Q}$ such that K, L contain each a unique prime lying over $p$ but KL does not. Another, different, triplet such that ...
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29 views

$Q\subset L$ with $G := \text{Gal}(L/Q)$, Is $L$ contained in the field of constructible numbers? [closed]

$Q \subset L$ is a finite Galois extension with $G := \text{Gal}(L/Q)$ and $G$ is isomorphic to $S_3$, the symmetric group on $3$ elements. Is $L$ contained in the field of constructible numbers?
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Feedback is needed: Describe the Galois group of the polynomial $x^5 -3 ∈ \mathbb{Q}(\zeta)[x]$ over $\mathbb{Q}(\zeta)$.

Describe the Galois group of the polynomial $x^5 -3 ∈ \mathbb{Q}(\zeta)[x]$ over $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive fifth root of unity. Here is what I attempted: Since the splitting ...
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1answer
61 views

How can I prove $f=x^n-a \in F[X]$ is separable?

$F$ is a field and we have $0\neq n\geq 2$. I need to show that $f=x^n-a$ is separable, meaning all its irreducible factors are separable. I first tried induction, but without avail. I couldn't ...
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26 views

Show that the Galois group is isomorphic to a dihedral group of order $8$.

Question. Let $f$ be the minimal polynomial of $\sqrt{1+\sqrt{7}}$ over $\Bbb Q$. Then, let $L$ be its splitting field. Show that $\text{Gal}(L/\Bbb Q)$ is isomorphic to a dihedral group of order $8$. ...
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Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?

We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. ...
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1answer
52 views

Unit roots group is isomorphic to $\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right]$ in a field of characteristic $p\ge0$

Let $K$ be a field so that the group of all unit roots of all orders $\mu_\infty=\bigcup_n {\mu_n}$ (where $\mu_n=\{x\in K\mid x^n=1\}$) splits on $K$. If $K$ is of characteristic $0$, take $p=1$; ...
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1answer
46 views

Intuition for equivalent definitions of Galois extension

Assume $K/F$ is a finite field extension. Then the following are equivalent: $|\operatorname{Gal}(K/F)| = [K:F]$. The fixed field of $\operatorname{Gal}(K/F)$ is $F$. $K$ is the splitting field of ...
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2answers
82 views

Finding Galois group of a separable polynomial

The question asks the following: Let $E$ be the splitting field of $x^4-10x^2-20$ over $\mathbb{Q}$. Find $Gal(E/\mathbb{Q})$. Since this polynomial is irreducible by Eisenstein's criterion, we ...
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41 views

Is $\operatorname{Gal}(\overline{\mathbb{F}_p} \,/\, \mathbb{F}_p)$ cyclic?

I know if $L/K$ is a extension of finite fields then $\text{Gal}(L/K)=\langle \phi \rangle$ where $\phi:L\longrightarrow L$ is the Frobenius automorphism. How can I show that it is also true for the ...
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1answer
23 views

Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$

Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ ...
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1answer
33 views

Degree of extension $\mathbb{C}/K$, where $K$ is maximal with the property $\sqrt{2} \notin K$

This question has been asked before but not really answered, but my query is a bit separate. To summarise the details: $K$ is a field maximal with respect to the property $\sqrt{2}\notin K$, any ...
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47 views

Dummit & Foote 14.4.4

Above I have posted the question (and Exercise 4.1.9 which the question references in its hint). I am a bit confused by the hint. It references the Galois group of L over F, but L is just a splitting ...
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30 views

Degree of an exclusionary Field Extension

Let's say I've got a field $\mathbb{Q}[i]$\ $\mathbb{Q}$. What's the degree of the field extension $\mathbb{Q}[i]$\ $\mathbb{Q}$ : $\mathbb{Q}$? Clearly without the exclusion this has a degree of 2; ...
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1answer
45 views

Why does an element of the Galois group of $K(\mu_n)/K$ maps an $n$-th root of unity to another $n$-th root of unity?

I am working through Keith Conrad's article on Cyclotomic extensions and have a question regarding the proof in Lemma 2.1. Let $K$ be any field and $\mu_n \subseteq K^\times$ be the multiplicative ...
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Is every finite Galois extension of $\mathbb{Q}_p$ which contains an $n$-th root of unity (where $\gcd(n,p)=1$) also abelian?

Let $K = \mathbb{Q}_p$ and $L$ be an arbitrary Galois extension of $K$ which contains an $n$-th root of unity (where $n$ is a natural number satisfying $\gcd(n,p)=1$). Question: Is $L/K$ abelian (...
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1answer
77 views

Why this polynomial reducible? (composite field)

In galois field of prime 2, in composite field $GF((({2}^2)^2)^2)$, There are irreducible polynomials and reducible polynomials. $GF(2^2):Q_1(x) = x^2+x+1,$ $GF((2^2)^2):Q_2(x) = x^2+x+\phi,$ $\...
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1answer
34 views

Galois group of field extension

I was asked to find the Galois group of the extension $\mathbb{Q}(\sqrt[3]{2},\sqrt{2},e^{\frac{2\pi i}{3}})$. Since the degree of the minimal polynomials of $\sqrt[3]{2},\sqrt{2}$ and $e^{\frac{2\pi ...
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1answer
50 views

Finding a minimal polynomial of a root of unity over a field extension

I'm trying to find the minimal polynomial of the seventh root of unity over the field $Q(i\sqrt{7})$. I know how to do this over the rationals and have proceeded to finding that $(x-1)(x^6 +x^5 +x^4 +...
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1answer
31 views

How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$

This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$. After reading this, I am still confused on how to find ...
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1answer
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Galois Group of K/Q and Embeddings K --> C

If any, what is the relation between $\text{Gal}(K/\mathbb{Q})$ and the set of embeddings (say $E$) of $K \to \mathbb{C}$? I ask this for two reasons: (1) The orders of $E$ and $\text{Gal}(K/\mathbb{...
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Is the set of constructible numbers in $\mathbb{C}$ algebraic extension of $\mathbb{Q}$?

Is the set of constructible numbers in $\mathbb{C}$ algebraic extension of $\mathbb{Q}$? I tried assuming that $z=a+ib$ and I am left to show that $ib$ is algebraic over $\mathbb{Q}$ whenever $b$ is. ...
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Relating Galois group of evenized-reciprocalized version to original Galois group

Let $Q\in{\mathbb Z}[X], Q=\sum_{k=0}^m q_kX^k$, with degree $m$ and Galois group $S_m$ (over $\mathbb Q$). Consider the evenized-reciprocalized polynomial $P(X)=X^{2m}Q\bigg(X^2+\frac{1}{X^2}\bigg)$. ...
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55 views

Galois group of $4X^{4m}+X^{4m-2}+X^2+4$ (non-abelian with roots of modulus one)

For $m\geq 2$, let $P_m=4X^{4m}+X^{4m-2}+X^2+4$. I have checked that for $1\leq m \leq 10$, 1) $P_m$ is irreducible over $\mathbb Q$ 2) All its roots have modulus $1$ 3) Its Galois group has ...
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1answer
20 views

Showing extension ring of a field is an integral domain

Let $F \subset K$ both be fields and $S \subset K$ be a non-empty set of $K.$ How do I show that $F[S]$ is an integral domain? The case for finite order $S$ is easy because $F[S]$ is just the image of ...
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17 views

Divide any given angle into $n$ equal parts using paper folding

I have known that it is possible to trisect an arbitrary angle by Hisashi Abe's method. Of course, it is easy to divide an angle into two or four equal parts using paper folding. My question is ...
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1answer
44 views

If $K(\zeta,\beta)/K$ is the splitting field for $f$, what can we say about $K(\zeta,\beta)/K(\zeta)$?

Problem setup: Let $n\in\mathbb{N}$ and let $K$ be a field whose characteristic does not divide $n$. The splitting field of $f=x^n-c$ ($c\not=0$) over $K$ is $K(\zeta,\beta)$ where $\zeta$ is a ...
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2answers
77 views

Find Gal$(f)$, here $f$ is the minimal polynomial of $x$ over $\Bbb{Q}(x^6)$

Let $f$ be the minimal polynomial of $x\in \Bbb{Q}(x)$ over $\Bbb{Q}(x^6)$. How to find the Galois group of $f$? Here is my thought, $f(t)=t^6-x^6$ is the minimal polynomial. Factor it out, we can ...
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3answers
69 views

Galois group of $x^p-a$ over $\mathbb{Q}$

I have found that the Galois group $G$ of $f=x^p-a$ over $\mathbb{Q}$ is of order $p(p-1)$. I need to show that if $P$ is a subgroup of $G$ of order $p$, then $P$ is normal and $G/P$ is cyclic. ...

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