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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$\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n \notin \Bbb{Q}(ζ_{11})$ for all positive integer $n$
I want to prove $\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n$ does not lie in $\Bbb{Q}(ζ_{11})$ for all positive integer $n$.
This problem arises from arithmetic geometry, but this problem ...
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Question about section 9.1 of Washington's "cyclotomic fields"
I am trying to understand the Basic Argument from Chapter 9.1 of Washington's "cyclotomic fields", and I can understand all but one part. Under Assumption 1: $p \nmid h^{+}(\mathbb{Q}(\...
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Any normal third degree extension of $\mathbb Q$ contained in $\mathbb C$ must be contained in $\mathbb R$
We let $K\subset \mathbb C$ be a finite normal field extension of $\mathbb Q$ such that $[K:\mathbb Q]=3$. Show that $K\subset \mathbb R$.
I know that no cyclotomic extension of $\mathbb Q$ will do ...
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On cyclotomics modulo prime powers
Regarding the ring $\mathbb{Z}[X]/\Phi_n(X)$ for the $n$'th cyclotomic polynomial $\Phi_n$, I find much literature on the nature of this ring taken modulo a prime $q$, that is literature on the ...
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The splitting field of $x^{20}-1$ over $\mathbb Q$
I am trying to find the splitting field of $x^{20}-1$ over $\mathbb Q$. I know that it has degree $8$ over $\mathbb Q$, and that $i$ and $\sqrt5$ are in the splitting field. I am also suspecting that $...
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Does $\mathbb{Z}[\zeta_n]$ contain $|x|=1$ with $x$ not a power of $\zeta_n$?
Let $\zeta_n=e^\frac{{2i\pi}}{n}$. Suppose $n$ is even. Then $\mathbb{Z}[\zeta_n]$ contains no roots of unity that are not powers of $\zeta_n$ because this is true of $\mathbb{Q}[\zeta_n]$ which is a ...
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$\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$
I want to prove $\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$ using the hint from "Introduction to Cyclotomic Fields" Exercise 2.1. $\zeta_{23}$ is the primitive 23-th root of ...
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Prove $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$
I am trying to prove that $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$, where $\xi_p$ is a primitive $p$-root of unity.
I am not sure how to do this. I have calculated the norm of $1-\xi_p$ and I ...
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$\sigma(\overline{z})=\overline{\sigma(z)}$ with $\sigma$ $\mathbb{Q}$-automorphism
I am trying to prove the following:
For fields $\mathbb{Q}\subset F \subset \mathbb{Q}(\xi_n)$, with $\xi_n$ a primitive $n$-th root of unity, prove that $\sigma(\overline{z})=\overline{\sigma(z)}$ ...
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Characterization of subfields of cyclotomic fields
The Kronecker-Weber theorem says that every finite abelian extension of $\mathbb{Q}$ lives in a cyclotomic one.
For what I know there could be some other.
Hence the natural question : is there a ...
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How to prove $x^p+y^p=\displaystyle{\prod_{k=0}^{p-1} (x+\xi_p^ky)}$
I am reading a "proof" of Fermat's last theorem which uses the assumption that $\mathbb{Z}[\xi_p]$ is a UFD (which is actually not true in general). This proof uses the following ...
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Can every algebraic number be written in terms of roots of unity?
Let $ \alpha $ be the root of some polynomial with integer coefficients. Can $ \alpha $ always be written as an algebraic expression using only rational number and roots of unity?
This is equivalent ...
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$\xi_p^i-\xi_p^j$ divides $p$ in the ring $\mathbb{Z}[\xi_p]$
I am trying to prove the following:
Let $p \geq 5$ be a prime and let $\xi_p\in \mathbb{C}$ be a primitive $p$-th root of unity (that is, $\xi_p\neq 1$ and $\xi_p^p=1$). I want to show that $\xi_p^i-\...
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$[L:K]$ is a Galois extention with Galois group G. Describe a map G to $\Bbb Z/p\Bbb Z$
Suppose K is a field containing p th roots of unity. Suppose $[L:K]$ is a Galois extention with Galois group $G$. Let $ a \in L$ not in $K$ s.t $a^p \in K$. Describe a nontrivial map $G \to \Bbb Z/p\...
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Parity of the class number of cyclotomic fields
I am interested if there are any heuristics on the parity of the class number of $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive root of unity.
Is it true that it is odd infinitely many often? Is ...
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Coprime ideals and Dedekind zeta function over cyclotomic fields
For a positive integer $m$, the $m$-th cyclotomic ring is $R = \mathbb{Z}[\zeta_m]$, the ring extension of the integers $\mathbb{Z}$ obtained by adjoining an element $\zeta_m$ having multiplicative ...
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Primes above $2$ in cyclotomic fields
I would be interested to know if there are any heuristics for how often it is the case that there is exactly one prime above $2$ in $\mathbb{Q}(\zeta_p)$, where $\zeta_p$ is a primitive $p^\text{th}$ ...
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Which $ \sqrt{n} $ are in a particular cyclotomic field
EDIT: Assume throughout that $ n $ is square free (thanks to Aphelli for pointing out that I need this). I also added the square free condition specifically to the first line for emphasis.
Suppose $ n ...
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Is the norm of a principal prime ideal principal?
Let $L/K$ be a finite extension of number fields. $\frak P$ denotes a principal prime ideal of $\mathcal O_L$. Then I want to show that $N_{L/K}(\mathfrak P)$ is principal. The norm is defined as $$ ...
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explicit generators for the unit group of a cyclotomic field
Let $p$ be an odd prime and $\zeta$ be a primitive $p$-th root of unity. Let $K^{+}=\mathbb{Q}(\zeta+\zeta^{-1})$ denote the maximal real subfield of $K=\mathbb{Q}(\zeta)$. For $2 \leq j \leq (p-1)/2$,...
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How to prove that: Automorphism $\zeta_n\to\zeta_n^p$ leaves prime ideal factor of *p* unchanged. [closed]
Let's
(n,p)=1
$\zeta_n$ - n-root of unity
$\mathfrak p$ - prime ideal factor of prime number p
How to prove that automorphism $\zeta_n\to\zeta_n^p$ leaves $\mathfrak p$ unchanged.
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If $P(P(x))=x^{36}$, does it imply $P(x)=x^6$?
Intro: This question has been inspired by the question If $P(P(x)-1) = 1 + x^{36}$ then $P(2)=$ . Here, we will consider (what was not explicitly mentioned in the original question) that $P$ is a ...
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"Easy" proof that $\Phi_n$ has degree $\phi_n$
Let $\Phi_n(x)$ denote the minimal polynomial of $\zeta_n = e^{2\pi i/n}$ over $\mathbb{Q}$. Now it's clear that $\Phi_n$ is irreducible, so the difficulty is showing that it has degree $\phi(n)$, ...
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The units in some extensions of integers
What are the units in the ring $\mathbb{Z}[a]$, where $a$ is the n-th root of unity in the complex field $\mathbb{C}$?
Katy
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Problem involving a cyclotomic extension
Let $\zeta_n := e^{2\pi i/n}$ and $K_n := \mathbb Q(\zeta_n) \cap \mathbb R$.
Suppose $\alpha \in K_n$ satisfies $\alpha^m \in \mathbb Q$ for some $m \geq 1$ such that $\alpha^j \notin \mathbb Q$ for ...
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When are the roots of cyclotomic units integers?
Let $p$ be an odd prime and $ζ=e^{\frac{2πi}{p}}$. Consider $ℤ[ζ]$, the ring of integers of the cyclotomic field $ℚ(ζ)$.
I want to ask the following question:
When is it possible to take the $n$-th ...
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Radicals in a Cyclotomic Field Extension
I was looking for the theorems which describe about all the kinds of radicals contained in a cyclotomic extension. With radical I mean the number, say $x$, is not in $\mathbb{Q}$ but some power, say $...
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Find $a \in \mathbb C$ such that $\mathbb Q (\zeta_5)= \mathbb Q(\sqrt 5, a)$ and $\left[ \mathbb Q(a): \mathbb Q \right]=2 $.
Let $\zeta_5= e^{\frac{2}{5} \pi i}= \dfrac{1}{4}(\sqrt 5 -1)+ i \left( \sqrt{\dfrac{5}{8}+ \dfrac{\sqrt 5}{8}} \right)$ be the principal fifth root of the unity. I know that $\left[\mathbb Q (\zeta_5)...
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Cyclotomic Polynomial Proof
I am trying to understand this proof for the nth cyclotomic polynomial $\Phi_n(x)$ having integer coefficients, given in Fraleigh’s 7th edition A First Course in Abstract Algebra. The proof assumes $K$...
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Subgroups for Galois group of cyclotomic etension
My teacher gave me this problem as a personal homework.
I need to determine if $\cos(2\pi/13)$ and $\cos(2\pi/55)\in K = \mathbb{Q}[\cos(2\pi/37),\cos(2\pi/15),\cos(2\pi/11)]$.
For $\cos(2\pi/13$) I ...
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On some congruence in cyclotomic field
Let $k = \mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field and $p$ a prime number verify $p\equiv 1 \pmod 5$.
Let $\lambda = 1-\zeta_5$ the unique prime of $k$ above $5$, precisely $5 = (1-\zeta_5)^...
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Factoring cyclotomic polynomials in tree form
$\newcommand\ord{\text{ord}_n(q)}$
I've got the following formula for the $n$'th cyclotomic polynomial over the field $\mathbb{F}_q$ ($q=p^r$ and $(n,p)=1$) assuming $q\equiv1(\bmod m)$ as well as ...
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Show that $1+\zeta+\dots+\zeta^{k-1}$ is a unit [duplicate]
Let $\zeta=e^{\frac{2\pi i}{p}}$ where $p\geq3$ is a prime. Consider the algebraic number field $K=\mathbb{Q}(\zeta)$. Let $k$ be a positive integer such that $(k,p)=1$ i.e., $k$ is co-prime with $p$. ...
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Identities in maximal real subfields of cyclotomic fields
For the following, let $m=2^{r_{0}}p_{1}^{r_{1}}\cdots p_{n}^{r_{n}}\geq 10$ be an even integer (so that $r_{0}\geq 1$), and let $\omega=e^{2\pi i/m}$ be a primitive $m$-th root of unity. Furthermore, ...
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Quintic residue symbole in the $5^{th}$ cyclotomic field
Let $p_1$ and $p_2$ be prime numbers such that $p_1 \equiv p_2 \equiv \pm2\bmod 5$. We have that $p_1$ and $p_2$ remain inert in the cyclotomic field $F = \mathbb{Q}(\zeta_5)$, where $\zeta_5$ is a ...
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Is it true that $\mathbb{Z}[\zeta_8]/\langle 1+3\sqrt{i}\rangle\cong\mathbb{Z}_{82}$?
I've been thinking a lot about ideals and factor rings, and I came upon the following. First, consider $\mathbb{Z}[\zeta_8]$, which is the cyclotomic ring of integers such that all $z\in\mathbb{Z}[\...
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Algebraic Integers in Cyclotomic fields
Let $\zeta$ be a primitive 8th root of unity in $\mathbb{C}$ and let $\alpha = \frac{1-\zeta}{2}$
(i) Determine the minimal polynomial of $\alpha$ over $\mathbb{Q}$
(ii) Is $\alpha^{-1}$ an algebraic ...
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Material for a first course on cyclotomic fields
I am aware that the two excellent graduate level books on theory of cyclotomic fields are:
1.Introduction to Cyclotomic Fields by Lawrence C. Washington
2.Cyclotomic Fields I and II by Serge Lang
Is ...
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Prove $1-ω$ is prime in $\mathbb{Z}[ω]$
Let $p$ be an odd prime and $ω=e^{2\pi i /p}$. Determine if $1-ω$ is prime in $\mathbb{Z}[ω]=\mathbb{Z}+\mathbb{Z}ω+\mathbb{Z}ω^2+...+\mathbb{Z}ω^{p-2}$
My attempt
I have tried using the definition of ...
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Sub-Fields of $Q(\zeta_{13})$ with a single generator.
Let $K=Q(\zeta_{13})$ a Number field of degree $12$ over $Q$ which is cyclic and has unique sub-fields $K_2,K_3,K_4$ and $K_6$ of degree $2,3,4$ and $6$ respectively. I need a single generator $\beta\...
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What are the integers that lack unique factorization in $\mathbb Z[\zeta_n],$ $n= 23$? [closed]
In the cyclotomic integers $\mathbb{Z}[\zeta_n]$, $n=23$ has class number $3$ and so unique factorization fails (https://en.wikipedia.org/wiki/Cyclotomic_field ). Can anyone give me examples of such ...
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Finding sine in the real subfield of a cyclotomic field
Consider the cyclotomic field $K=\mathbb{Q}(\zeta)$ where $\zeta = \exp(2\pi i/n)$ is a primitive $n$-th root of unity. I think it's pretty well known that the maximal totally real subfield of $K$ is $...
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How to write an arbitrary element of cyclotomic extension of 6th root of unity?
Let's say $\omega$ is a primitive $6^{\rm th}$ root of unity. Then we know $[\mathbb{Q}(\omega):\mathbb{Q}]=\varphi(6)=2$. If I count the number of linearly independent basis elements for $\mathbb{Q}(\...
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When can a representation of a finite group be defined over certain cyclotomic fields?
Let $G$ be a finite group and $\rho \colon G \to \mathrm{GL}(n, \overline{\mathbb Q})$ be a representation. A theorem of Frobenius says that $\rho(G)$ is conjugate (in $\mathrm{GL}(n, \overline{\...
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How to see that the cyclotomic extension is a separable extension?
$\mathbb{F} \subset \mathbb{E}$ is a cyclotomic extension, if
$\mathbb{E}$ is the splitting field of $x^n-1$ over $\mathbb{F}$.
We were directly told that $\mathbb{E} \mid \mathbb{F}$ is a Galois ...
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Prove that $N(a+b\xi)=\frac{a^5+b^5}{a+b}$
Let $\xi=e^{\frac{2\pi i}{5}}$, and let $K=\mathbb{Q}(\alpha)$ be a number field. I want to prove that $N(a+b\xi)=\frac{a^5+b^5}{a+b}$, where $a+b\neq0$.
If I define the $4$ inmersion like this: $\...
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Maximal abelian $p$-extension unramified away from $p$
I am having trouble seeing how class field theory implies the following isomorphism. Let $F_n=\mathbb{Q}(\mu_{p^{n+1}})$, let $L_p/F_n$ be the maximal unramified abelian $p$-extension (i.e. the $p$-...
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Dedekind zeta function for a subfield of cyclotomic field
Can someone give me a proof or a reference of a proof of Proposition 23 here. I cite it below. $\mu_n$ is an $n$th root of unity.
Proposition 23. Let $F$ be a number field contained in $\mathbb{Q}\...
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Upper bound for the cyclotomic number (0,0) of general order over characteristic-2 finite field.
$\mathbb{F}_q$ is a finite field with characteristic 2, namely, $q=2^n$ is a power of 2. $g$ is a generator of the multiplicative group $\mathbb{F}_q^\times=\mathbb{F}_q\backslash\{0\}$.
Cyclotomic ...
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The number of solutions to $a+b=1$ in a multiplicative subgroup of a finite field $GF(q)$
I have been wondering what is the number of solutions to $a+b=1$ in an arbitrary multiplicative subgroup $H$ (order $r$) of some finite field $GF(q)$, where $q=p^n$ is a prime power and $a, b \in H$. ...
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