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Questions tagged [cyclotomic-fields]

Cyclotomic fields are fields where a primitive root of unity is added to the rational numbers. These fields are common in algebraic number theory.

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Find all subfields of $\mathbb{Q}(\mu_{24})$

Problem: Let $\mu_{24} \in\mathbb{C}$ be a primitive 24'th root of unity and let $L = \mathbb{Q}(\mu_{24})$ be the 24'th cyclotomic extension of $\mathbb{Q}$. List all subfields of $L$ in the form $\...
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Chebotarev's theorem on cyclotomic extensions.

Let $K=\mathbb{Q}$ and $L=\mathbb{Q(\zeta_n)}$, where $\zeta_n$ is the $n$-th primitive power of unity. If $C$ is a conjugacy class of $G=\text{Gal}(L/K)$, Chebotarev's density theorem says that the ...
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Are Galois conjugates of a prime of a cyclotomic ring also primes?

For the sake of simplicity consider $\mathbb{Q}[\zeta_{5}]$. If a cyclotomic integer $z\in\mathbb{Z}[\zeta_{5}]$ is a prime of the integer ring, is it true that its Galois conjugates $\{z, \sigma_1(z),...
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Questions on $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$

a. Determine $[\Bbb Q(\zeta_7 + {\zeta_7}^{-1}):\Bbb Q]$ . b. Is $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$ a Galois extension? c. Is $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$ a radical ...
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Is $\Bbb Q(\sqrt{15}) \subset \Bbb Q(\zeta_{15})$?

Is $\Bbb Q(\sqrt{15}) \subset \Bbb Q(\zeta_{15})$ ? $Gal(\Bbb Q(\zeta_{15}):\Bbb Q)=\varphi(15)=8$ . Thus $[\Bbb Q(\zeta_{15}) \cap \Bbb R : \Bbb Q]=\frac{\varphi(15)}{2}=4$, so it does not quite ...
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31 views

is $\Bbb Q(2^{1/4}) \subseteq \Bbb Q(\zeta_{16})$

Is $\Bbb Q(2^{1/4}) \subseteq \Bbb Q(\zeta_{16})?$ $\Bbb Q(2^{1/4}) \subseteq \Bbb Q(\zeta_{16})\implies x^4 - 2 $ splits in $\Bbb Q(\zeta_{16})\implies \Bbb Q(2^{1/4},i)\subseteq \Bbb Q(\zeta_{16})\...
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Dedekind theorem of ramification in cyclotomic fields

Let $\Bbb Q(w)$ denote the $n$-th cyclotomic field then$\Bbb Z[w]$ is its ring of integers, $d$ denotes the discriminant of $\Bbb Q(w)$, and $p\mid d$ then $p\Bbb Z$ must ramify in $\Bbb Q(w)$. In ...
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1answer
36 views

Confusion about Quadratic Fields

I am told that $K_p = \frac{\mathbb{Q}[x]}{(\Phi_p(x))}$ where $\Phi_p(x) = x^{p-1} + ... + x + 1 $. Subfield L is then defined such that: $\mathbb{Q} \subset L \subset K_p$ where $[L:\mathbb{Q}] = ...
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Inertia fields and decomposition fields in cyclotomic extensions

I've been trying to figure out the following problem. Given a prime $p$, and $\zeta_m = e^{2\pi i /m}$, and the factorization $m = p^kn$ with $(p,n) = 1$, then we know that the Galois group of $\...
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Characterize the generators of the totally real subfield of an arbitrary cyclotomic field

Suppose that K = $\mathbb{Q}(\zeta )$ where $\zeta $ is a primitive nth root of unity, then the matching totally real subfield is ${{\text{K}}^{+}}$ = $\mathbb{Q}(\zeta +\text{ }1/\zeta )$ . (These ...
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Question regarding Galois group and algebraic extensions

From Dummit and Foote I was reading about cyclotomic extensions, where I came across the definition of algebraic extension saying the extension $K/F$ is an algebraic extension if $K/F$ is Galois and $...
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60 views

Find Galois group of $L:\mathbb{Q}(\sqrt{-5})$ where $L=\mathbb{Q}(\zeta_{20})$

I would like to find the Galois group of the field extension $L:\mathbb{Q}(\sqrt{-5})$, where $L=\mathbb{Q}(\zeta_{20})$ and $\zeta_{20}$ is a primitive 20th root of unity (in $\mathbb{C}$). We know ...
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Minimal polynomials of primitive elements compared to normal elements

Let $gcd(n,q)= 1$ Consider $x^n - 1 \in \mathbb{F}_{q}[x]$, and let $\mathbb{F}_{q^t}$ be a splitting field for $x^n - 1$ over $\mathbb{F}_q$. Then, $\mathbb{F}_{q^t}$ contains a primitive $n^{th}$ ...
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What is the tensor product $\mathcal{O}_K \otimes_\mathbb{Q} \mathbb{R}$?

Let $m \in \mathbb{N}^*$. Denote the $m$-th cyclotomic polynomial by $\Phi(x)$ and a complex primitive $m$-th root of unity by $\omega$. Let $K = \mathbb{Q}[x]/\langle \Phi(x) \rangle$ (which is ...
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How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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31 views

A quadratic cyclotomic extension

Let $\zeta_n$ be a primitive $n$-th root of unity with $n > 2$. How to show (supposedly using Galois theory) that the extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}(\zeta_n+\zeta_n^{-1})$ has degree $2$...
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70 views

Quadratic polynomial satisfied by $\zeta_5+\zeta_5^{-1}$

I got one problem from Dummit Foote stating that determine the quadratic polynomial satisfied by the period $\alpha=\zeta_5+\zeta_5^{-1}$ of the the $5th$ root of unity $\zeta_5$. Determine the ...
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47 views

Let $K=\Bbb Q (e^{\frac{2 \pi i}{7}})$ and $\alpha \in K - \Bbb Q$. Then show that, $\Bbb Q(\alpha)=K$

Let $K=\Bbb Q (e^{\frac{2 \pi i}{7}})$ and $\alpha \in K - \Bbb Q$. Then show that, $\Bbb Q(\alpha)=K$ . My attempt: I have computed the minimal polynomial of$e^{\frac{2 \pi i}{7}}$ over $\Bbb Q$ to ...
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Finding the Normal Basis of Cyclotomic field

So let $p$ be a prime number and $\zeta_p$ the p-th roots of unity. I want to proof that $ B = \{ \zeta_p, \zeta_{p}^{2}, \dots, \zeta_{p}^{p-1} \}$ is the normal basis of $\mathbb{Q}(\zeta_p)/\mathbb{...
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61 views

algebraic integer in $\mathbb{Q}$($\zeta_n$)

Let $\chi(\cdot)$ be an irreducible (over $\mathbb{C}$) character of an representation in a finite Group G: Show that $\chi(g)$ is an algebraic integer in the cyclotomic field $\mathbb{Q}$($\zeta_n$),...
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Density of integers represented by norm form (cyclotomic field)

Let $K=\mathbb Q(\zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ ...
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Isomorphism between two versions of $GF(2^3)$ [duplicate]

I have $GF(2^3)$ generated by $\Pi_1(\alpha)=x^3+x+1$ and $GF(2^3)$ generated by $\Pi_2(\alpha)=x^3+x^2+1$. $\bullet$ $\Pi_1(\alpha)=x^3+x+1$ $000=0, 100=1,010=\alpha,001=\alpha^2,110=\alpha^3, 011=\...
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Cyclotomic cosets $\pmod 8$, $p=3$

First time dealing with cyclotomic cosets, and I've solved my first exercise; I wish to have any kind of corrections/hints you can write to me. Find all the cyclotomic cosets $\pmod 8$, $p=3$ ...
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55 views

On a special kind of six dimensional vector subspace of $\mathbb C^9$ related to the primitive $9$-th root of unity

Let $\mu=e^{2\pi i/9}$ . Let $u_j:=(\mu^j,\mu^{2j}, \mu ^{3j},...,\mu^{9j})^T \in \mathbb C^9$, for $j=1,...,9$. Let $V$ be the vector subspace of $\mathbb C^9$ spanned by $\{u_2,u_3,u_4,u_5,u_6,u_7\...
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$\chi(\cdot)$ is an algebraic integer in $\mathbb{Q}$($\zeta_n$)

Let $\chi(\cdot)$ be an irreducible (over $\mathbb{C}$) character of an representation in a finite Group G: Show that $\chi(g)$ is an algebraic integer in the cyclotomic field $\mathbb{Q}$($\zeta_n$),...
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Number of prime norms $q$ not exceeding $2^p$

Suppose $K=\mathbb{Q}(\zeta_p)$ is the $p$th cyclotomic field for some prime $p$. What is the approximation for the number of primes $q<2^p$ such that $q$ is the norm of some cyclotomic integer (or ...
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Understanding problem in Milne's notes on class field theory

I was going through Milne's notes on class field theory and approached the following difficulty in understanding: On page 155, after the discussion of the conductor, he just writes that any Abelian ...
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Cyclotomic cosets and quadratic residues

Let $p$ be a prime and let $Q$ be the set of quadratic residues and $N$ the set of nonresidues. Assume $2 \in Q$. When I look a the cyclotomic cosets mod $p$, I get ${\{0}\}, {\{Q}\}, {\{N}\}.$ For ...
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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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Expressing $\zeta^k+\zeta^{-k}$ as a polynomial in $\zeta+\zeta^{-1}$.

Let $\zeta$ be an $n$-th root of unity and let $\chi:=\zeta+\zeta^{-1}$. Then $\zeta^k+\zeta^{-k}=P_k(\chi)$ where $P_k\in\Bbb{Z}[X]$ is a polynomial not depending on $n$. For example we have \begin{...
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How to describe lattice of subfields of $\mathbb{Q}(\zeta_{13})$

I have to describe lattice of subfields of $\mathbb{Q}(\zeta_{13})$. I know that : $$[\mathbb{Q}(\zeta_{13}):\mathbb{Q}]=\phi(13)=12$$ And $$\zeta_{13}=e^{\frac{2\pi ik}{13}} $$ ...
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70 views

If $p\equiv1\pmod{4}$, do we have $\sqrt{p}\in\mathbb{Q}(\zeta_p)$?

Why is it true that if $p\equiv1\pmod{4}$ then $\sqrt{p}\in\mathbb{Q}(\zeta_p)$?
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Is $\{sin(n \pi/2N): n\in \mathbb{Z},0<n \leq N\}$ linearly independent over $\mathbb{Q}$

I want to consider the dimension of $\mathbb{Q}$ vector space generated by $\{sin(n \pi/2N):n\in \mathbb{Z}\}$ where $N$ is a fix integer. My intuition is that if $N$ is not divisible by 3 then all ...
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$\mathbb Q(\zeta_m)\cap\mathbb Q(\zeta_n)=\mathbb Q(\zeta_d)$

Prove that $\mathbb Q(\zeta_m)\cap\mathbb Q(\zeta_n)=\mathbb Q(\zeta_d)$ where $d=\gcd(m,n)$. I want to solve this problem without Galois theory. I know only about field extension. For example, ...
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The Stickelberger relation in Ireland&Rosen's Number Theory book

Could anyone please take a look at the very last expression at the very bottom of Ireland&Rosen page 214? I am refering to their A Classical Introduction to Modern Number Theory, 2nd Edition. The ...
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Primes $p=1\pmod 5$

Let $p=1\pmod 5$ be a prime. Then $p=x^4 + x^3z + x^2z^2 + 5xy^3 - 5xy^2z + xz^3 + 5y^4 - 10y^3z + 10y^2z^2 - 5yz^3 + z^4$ for integers $x, y,$ and $z$. Is there a proof for this? For instance ...
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Minkowski bound of $229$-th cyclotomic field

If $K = \mathbb{Q}\zeta_{229}$, the $229$-th cyclotomic field, Wwat is a good approximation for the bound $n$ such that every class in the ideal class group of $K$ contains an integral ideal of norm $...
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Minimal polynomial generators of aurifeuillan factors cyclotomic polynomials

In order to answer this question, I came up with a way to generate the reciprocals of aurifeuillan factors of $n$-th cyclotomic polynomials for odd prime $n$. If $n=1\pmod 4$, then $\Phi_n(nx^2)$ has ...
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Calculating class numbers

It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book Algebraic Number Theory: "Example 7.20 For $K=\mathbb{Q}(\...
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Is $h(n) < 2^n$ for all $n$? ($n$th cyclotomic field class number growth)

Is it ever the case that (for prime $n$) the $n$-th cyclotomic field class number, $h(n)$, is greater than $2^n$? (List of class numbers of $n$-th cyclotomic field for prime $n$). For instance, the $...
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class number for special cyclotomic fields

Are the class number of the cyclotomic fields $Q(\mu_n)$ known, where $n=2^k$ ? Especially for $k=12$ ? Such fields are often used in Post Quantum cryptography, e.g. in Ring Learning With Errors ...
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If $n$ divides $m$, prove that $\mathbb{Q}(\zeta_{n}) \subset \mathbb{Q}(\zeta_{m})$

If $n$ divides $m$, prove that $\mathbb{Q}(\zeta_{n}) \subset \mathbb{Q}(\zeta_{m})$. If $n$ divides $m$, so $m = nk$ and $\varphi(m) = \varphi(n)\varphi(k)(k,n)/\varphi((k,n))$, then $\varphi(n)$ ...
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134 views

Root of unity belongs to Z/qZ. How?

EDIT: Really sorry for not posting this initially.. maybe it's easier to understand now. Source, page 6. I've stubled upon a statement similar to this: "Let $m,q$ be two integers such that $\mathbb{...
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71 views

How to show that $\mathrm{Gal}(\Bbb Q(\xi_{p^n})/\Bbb Q(\xi_{p^m}))$ is cyclic?

How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{p^n})/\Bbb Q(\xi_{p^m})]$ cyclic? Here $p \not = 2$ is a prime number and $n>m>0$ are natural numbers. $\xi_{p^n}$ and $\xi_{p^m}$ are cyclotomic ...
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123 views

Is $\sqrt[3]5+\sqrt[3]7$ contained in a cyclotomic extension?

Is $\sqrt[3]5+\sqrt[3]7$ contained in a cyclotomic extension? I am trying to prove that $\Bbb Q(\sqrt[3]5+\sqrt[3]7)/ \Bbb Q $ is an abelian or non abelian extension, so I will prove that this is ...
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1answer
101 views

Question about the Cyclotomic polynomial and Cyclotomic field.

I am studying for my Algebra PHD Qualifying and I can not solve the first 3 parts of this problem. I think I got the 4 correct. I have some ideas but I can't put them together to solve it. Anyone can ...
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1answer
41 views

Cyclic units in $Z_p$ towers

Let $\mathbb{Q}_n$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension over the rationals, i.e. the unique real subfield of the cyclotomic field $\mathbb{Q}(\zeta_{p^{n+1}})$ of degree $...
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252 views

Closest cyclotomic integer to a cyclotomic number?

Let's take a cyclotomic field of the form $K=\mathbb{Q}(\zeta_n)$ where $\zeta_p$ is the $n$th root of unity. Then the ring of integers of $K$ is $\mathcal{O}_K= \mathbb{Z}(\zeta_n)$. Is there a ...
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82 views

About the $\mathbb{Z}$-ranks of groups of cyclotomic numbers

Let $m$ be a natural number and $\zeta_m$ be a primitive $m$th root of unity. Let $G_m$ be the galois group of $\mathbb{Q}(\zeta_m)$ over $\mathbb{Q}$. Let $V_m$ be the multiplicative group generated ...
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62 views

$\sin(k\pi/n)$ an integer multiple of $\sin(\pi/n)$

Is there any positive integers $(n,k) \neq (6,3)$ with $n\geq 6$ and $2\leq k\leq n/2$ such that: $$\frac{\sin(k\pi/n)}{\sin(\pi/n)} \in \mathbb{N} ?$$ For $n\leq 57$ the answer is no (by computation),...