# 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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### 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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### 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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### If a number defined by radicals is a root of a polynomial with integer coefficients, then so is the same number taking another branch of a root

Suppose x is a number written with an expression composed of rational constants, sums, subtractions, multiplications, division and extraction of n-th roots. That is, $$x=\alpha^{1/n}\beta+\gamma$$ ...
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### If $n$ is odd then $\prod_{j=1}^{\frac{n-1}{2}} (\zeta_n^{t j}+\zeta_n^{-t j}+2)=2^{(t,n)-1}$

Is this proof correct? (Notation: $\zeta_n=\text{exp} (2\pi i/n)$) If $n$ is odd then $\gcd (2,n)=1$. Hence, \begin{align*} \prod_{j=1}^{\frac{n-1}{2}} (\zeta_n^{t j}+\zeta_n^{-t j}+2)&=\prod_{j=1}...
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### Finding a square root in a cyclotomic field

Let $x\in\mathbb{Q}[\zeta_{p}]$. Suppose that $$x=a_0+a_1\zeta_p+...+a_{p-2}\zeta_p^{p-2},\quad a_i\in\mathbb{Q}$$ and that $x$ has a square root in $\mathbb{Q}[\zeta_{p}]$, i.e. there exists an ...
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### On the representation of $\sqrt{\pm p}$ in the integral basis of $\mathbb Q(\zeta_p)$

I took another look at my previous question on proving a certain trigonometric identity related to the braced heptagon: $$\sin\frac\pi7-\sin\frac{2\pi}7-\sin\frac{4\pi}7=-\frac{\sqrt7}2$$ Around the ...
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### Why is $\mathbb{Q}[\zeta_{p}] = \{a_{0}, a_{1}\zeta, \dots, a_{p-2}\zeta^{p-2}, a_{i} \in \mathbb{Q}\}$, where $p \geq 2$ is a prime?

Let $p$ be a prime number $\geq 2$ and $\zeta := \zeta_{p}$ a complex number where $\zeta \neq 1$ and $\zeta^p=1$, i.e., is a $p$th root of unity. This is a question from a brazillian book, Introdução ...
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### If $\alpha \in \mathbb Q[\zeta_n]$ satisfying $\alpha^m=2$ for some positive integer $m$, then $m=1$ or $2$.

Let $\zeta_n:=$ primitive $n$-th root of $1$ over $\mathbb Q$. And let $\alpha \in \mathbb Q[\zeta_n]$ satisfy $\alpha^m=2$ for some positive integer $m$. Then I have to show that $m=1$ or $2$. We ...
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### Ramification in cyclotomic extension

I am trying to understand how primes ramify in $\mathbb Q(\zeta _p)/\mathbb Q$ for $p$ prime. From class field theory, how a prime $q$ ramifies depends only on $q \mod p$. I have following particular ...
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### Is the absolute value of the sum of six of the 16th roots of unity ever a nonzero integer?

Let $\zeta_1, \ldots \zeta_{16}$ be the $16$th roots of unity. For the proper subset $J \subset \{1,2,\dots,16\}$ and $|J|=6$, can the following sum ever be satisfied for an integer not equal to zero? ...
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### How to prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$?

How do I prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$? I calculated some examples in Wolfram Alpha where the results were of this specific form, and it ...
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### On the quintic power residue symbol

Let $k = \mathbb{Q}(\zeta_5)$ the cyclotomic field, where $\zeta_5$ a primitive $5^{th}$ root of unity. let $p$ a prime integer such that $p \equiv 1 [5]$, more exactly $p \equiv 1 [25]$, then $p$ ...
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### show that there exists a unique subfield $H$ of $\mathbb{Q}(\xi_9)$ such that $[H:\mathbb{Q}]=2$.

Let $\xi_9 = e^{\frac{2\pi i}{9}}$, I want to show that there exists a unique subfield $H$ of $\mathbb{Q}(\xi_9)$ such that $[H:\mathbb{Q}]=2$. Is this correct? In that case How one can prove this? I ...
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### Quadratic subfields of the cyclotomic field $\mathbb{Q}(\zeta_{14})$

In a nutshell, my question is: what is degree of the field extension $\mathbb{Q} \, ( \zeta_{14} + \zeta_{14}^9 + \zeta_{14}^{11})$ over $\mathbb{Q}$? As to why I'm asking this, I was trying to ...
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I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the ...
### How do you find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$?
In the comments on a question I posted earlier it was recommended that I find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$. How should I go about this? What would be a good ...