# 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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### Galois Group of $\mathbb Q$ Surjects onto Certain Cyclotomic Extension

Let $\ell$ be a prime and let $G_{\mathbb Q, \ell}$ denote the Galois group of $\mathbb Q(\mu_{\ell^\infty})/\mathbb Q$, the extension of $\mathbb Q$ formed by adjoining all primitive $\ell^n$th roots ...
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### Linear Dependence of Primitive Roots of Unity

Consider the cyclotomic field $\mathbb{Q}(\zeta_n)$. We know that the set of primitive roots $\Pi_n=\{\zeta_n^m:(m,n)=1\}$ generates $\mathbb{Q}(\zeta_n)$ as a field. However, what happens when we ...
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### $\sqrt{p^*}$ contained in $\Bbb Q(\zeta_p)$ [duplicate]

Let $p$ be a prime. I read several times e.g. here that the square root $\sqrt{p^*}$ (where $p^*:=(-1)^{(p-1)/2}p$) is contained in $\Bbb Q(\zeta_p)$. Is there a "standard argument" see it? ...
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### Proving $\mathbb{Q}(\sqrt 2) \subset \mathbb{Q}(\zeta_n) \iff n = 8 \cdot m$ for $m \in \mathbb{N}$

My ultimate goal is showing that $\mathbb{Q}(\sqrt 2) \subset \mathbb{Q}(\zeta_n) \iff n = 8 \cdot m$ for $m \in \mathbb{N}$. I have showed that for $n= 8 \cdot m$ the result holds. There is only one ...
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### Kummer's Lemma and $1+\zeta$

In lecture we were told to think about the following: Kummer's Lemma: Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ ...
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### rational elliptic curve: torsion subgroup defined over abelian extensions

I'm thinking about a question, that is if E is a rational elliptic curve, K is a cyclotomic field, how can I find the minimal subfield of K such that the torsion subgroup over K can also be defined ...
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### How to compute the different ideal of the cyclotomic field extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$? [closed]

Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity. There is a notation called different ideal, which is defined to ...
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### What is known about the minimal absolute value in $\mathbb{Z}[\zeta_n]\setminus\{0\}$?
Here $\mathbb{Z}[\zeta_n]$ is the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the $n$th root of unity. In my mind $\mathbb{Z}[\zeta_n]$ looks like a grid in the ...