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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Closed formula for $\Phi'_n(\zeta)$ as complex number

This is a follow-up to this question. The one thing that did not get a completely satisfactory answer there is: If $\Phi_n$ denotes the $n$-th cyclotomic polynomial, and $\zeta^k_n = e^{2k\pi i/n}$ is ...
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More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=6m+1\tag1$$ and the cubic, $$x^3+x^2-2mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
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Summation involving Roots of Unity

Problem Let $x \in \mathbb{R_{\geq 0}}$, $\alpha \in (0,1]$ and $p \in \mathbb{N}$. Define $$S(x)=\sum_{k=0}^{p-1} \frac{(e^{\frac{2\pi k}{p}i})^{x+1}}{e^{\frac{2\pi k}{p}i}-\alpha}$$ Does there ...
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This question is based on this other question from @Malkoun, and more specifically the comments. Context. Let's recall the following proof which shows that $-1$ is a square if $\mathbb F_p$ for $p\... 0answers 22 views Is there a term for a set of eigenvalues not containing any roots of unity except 1? Is there a term describing a matrix$A \in \mathbb C^{n\times n}$whose eigenvalues are all either$1$or not a root of unity? Is there a term for a holomorphic selfmap$F$of$\mathbb C^n$fixing$0 ...
I'm working on a problem, but I haven't been able to get much out of it... Given that $\zeta$ is a primitive complex n-th root of unity, show that the complex numbers $1,\zeta,\zeta^2,...,\zeta^{n-1}$...