# 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 algebraic integers in $\mathbb{Q}[x]$ where x is a root of unity

prove that the set of algebraic integers in$\mathbb{Q}[\zeta_p]$ are exactly $\mathbb{Z}[\zeta_p]$where $\zeta_p$ is a primitive p-th root of unity(may assume p is prime) I'm sure that the problem is ...
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### Properties of a certain divisor of the $m^{\rm th}$ root of unity in the $m^{\rm th}$ cyclotomic ring.

I am an analyst who is new to algebraic number theory. I am learning on Cyclotomic polynomials and came across this problem, which I am having hard time solving. Let $\xi$ be an $m^{\rm th}$ complex ...
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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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### $K(\zeta)/k(\zeta)$ 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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### Exercise 4.8 from Marcus'book “Number fields”

I report here the Exercise with my sketch of a solution. (a) By Exercise 30 Chapter 3, I know that there are infinitely many primes $q \equiv 1 \bmod r$. Moreover, I know that the ring of integers of ...
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### 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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### Pseudo-isomorphism in Iwasawa Theory

Let $\Lambda$ denote the Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$. We know that $\Lambda$ is homeomorphic to $\mathbb{Z}_p[[T]]$, ring of power ...
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### Inertia group of an extension of Local Fields

I am going through a proof in Chapter V of Neukirch's Algebraic Number theory, and would require some clarifications. The theorem we are trying to prove is that the local version of the Kronecker-...
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### Roots of unity in $\mathcal{O}_{K}$, where $K=\mathbb{Q}(\sqrt{5},\sqrt{-2})$.

Let $K=\mathbb{Q}(\sqrt{5},\sqrt{-2})$. I already showed that the ring of integers in $K$ is $\mathbb{Z}(\frac{\sqrt{5}+1}{2},\sqrt{-2})$ and the discriminant is $1600$. Now the goal is to show that ...
In Washington's Introduction to Cyclotomic Fields, we are given the following lemma (1.4): The ideal $(1 - \zeta)$ is a prime ideal of $\mathscr{O}$ and $(1 - \zeta)^{p-1}$= (p). Therefore p is ...
Let $p$ be an odd prime, $\zeta_p=e^{{2\pi i}/p}\in\mathbb{C}$, $F=\mathbb{Q}(\zeta_p)$. Let $E\subseteq\mathbb{C}$ be the splitting field of $X^p-2$ over $\mathbb{Q}$. Let $F_0=F\cap\mathbb{R}$ ...