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

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Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
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Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
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For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed as ...
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Decomposition field of $x^n - 1$ over $Z_p$

In my textbook, it is said that (Primitive root) Let $p$ be a prime and $n > 1$ be a natural number. The set of all the roots $\alpha$ of the polynomial $x^n - 1 \in Z_p[x]$ forms a cyclic group ...
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Primitive $p$-th root of unity with characteristic $p$

I struggle on this since two days, and still found no answer. My course states the following: If the characteristic of $K \neq p$, then they are exactly $p-1$ different $p$-th roots of unity in the ...
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maximum distance between n points on unit circle

Let's n points $M_k$ on the unit circle. Find the position of these points such that $$P_n = \prod_{1 \le k \le l \le n}M_kM_l$$ (product of all the distances between the points) is maximum. With the ...
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Complex Endomorphism $f$ with orthogonal Eigenspaces implies that $f$ is unitary

I already posted this question earlier today without much success. After 3 hours and only 10 views I decided to give it another shot :) Of course I'll delete the second post! Let $V \ne 0$ be an ...
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Property of roots of unity

Let $R$ be a commutative ring and $\zeta$ be a root of unity in $R$ such that $\mathrm{ord}(\zeta) = n$. If $n = p_1^{e_1} \dots p_r^{e_r}$ is the decomposition in prime factors of $n$, is it true ...
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Closed-Form Expression for a Certain Sum Over Roots of Unity

Let $p$ be an odd integer $\geq3$, let $n$ be an arbitrary positive integer, let $c$ be a non-zero complex number, and let $\omega$ be a complex variable. Does anyone know of a closed-form expression ...