Questions tagged [roots-of-unity]
numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field
1,077
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Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$
Let $c_k(n)$ denote Ramanujan's sum, and $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Prove that
$$\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$$
My attempt was to ...
- 702
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$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity
I need to prove/disprove the following statement:
Every algebraically closed field $K$ contains a 10th root of unity.
I don't think the statement is true. My counterexample is as follows:
Let's take ...
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Natural map of automorphism groups
Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
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Resources on infinite series involving $n$'th roots of unity
Background
I was wondering whether there any books, articles, or other in-depth treatments of infinite series involving the roots of unity. Let $\omega_{n} := e^{2 \pi i / n}$ be the $n$'th root of ...
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Help understanding derivation of identity $\sum_{k=1}^{n}\cot^4\left({k\pi\over 2n+1}\right)=\frac{1}{45}n(2n-1)(4n^2+10n-9)$
This question regards understanding some of the steps in the derivation of the identity for $\sum_{k=1}^{n}\cot^4\left({k\pi\over 2n+1}\right)$
It is shown 1 that using Vieta's formula that
$\sum_{k=1}...
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Can we find the root of this equation
Give an equation below:
$$
\frac{x^k-a^k}{x-a}=c \qquad (1)
$$
where $1<a<x$, $0<k<1$, and $c>0$.
I can easily find the numerical root of (1) by using Newton's method or the other tools....
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Generalization of Integer-Powered Sums Problem
I am trying to solve a problem that involves the sum of the $n$-th roots of positive reals.
Specifically, the task is to determine all sets of positive reals $a_1, a_2, a_3$ such that $\sqrt[n]{a_1}+\...
- 1,023
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Understanding roots of unity in quadratic fields
Suppose we have a quadratic field $\mathbb{Q}(z)$ with $z \in \mathbb{C} \setminus \mathbb{Z}$ and $z^2 \in \mathbb{Z}$. How would one go about determining the possible $n^{\text{th}}$roots of unity ...
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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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Certain sum involving roots of unity like the Lambert series
Let $\ell$ be an odd prime $, \zeta=e^{\frac{2 \pi i}{\ell}}, v \in \mathbb{Z}_{>0}$. Then how to prove there exist an integer $N \equiv -v \ (\bmod \ell)$,
$$
\begin{aligned}
\frac{\zeta^v}{(1-\...
- 1,016
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Anything interesting known about this generalization of even and odd functions?
Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$,
$$f(\omega z) =...
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Degree of field extension $\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q$
$n>2$ is an odd squarefree integer.
Let $\zeta_n=\mathrm{e}^{i\frac{2\pi}n}$ be a primitive $n$-th root of unity.
Is it true that $[\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q]=\...
- 3,047
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The unit digit of $\prod_{k=0}^{97}\left(2+\alpha_{k}^2\right)$, where $\alpha_0,\alpha_1,....,\alpha_{97}$ are the $98^{th}$ roots of unity
The unit digit of $$\prod_{k=0}^{97}\left(2+\alpha_{k}^2\right)$$, where $\alpha_0,\alpha_1,....,\alpha_{97}$ are the $98^{th}$ roots of unity
My Approach: Since $\alpha_0,\alpha_1,....,\alpha_{97}$ ...
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Summation of reciprocals of nth roots of unity
I let the term inside the summation be equal to x, put $a$ in terms of x and used $a^n$=1 but because I have not done binomial yet I got stuck at that point. I was thinking of opening using binomial ...
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Let the value of $\log_{2} \left[\prod_{a=1}^{2015} \prod_{b=1}^{2015} \left(1+e^{\frac{2 \pi i a b}{2015}}\right)\right]$
Let the value of $$\log_{2} \left[\prod_{a=1}^{2015} \prod_{b=1}^{2015} \left(1+e^{\tfrac{2 \pi i a b}{2015}}\right)\right]$$ is $N$, then which of the following is/are true
(a) $N$ is divisible by $5$...
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