# 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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### 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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### 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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### 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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### 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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### 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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