# 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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### Permutating the coeffecients of equations with roots of unity.

I"ve noticed the roots of $$(x^k-1 =0) = (1-x^k=0)$$ This is simple enough since changing +/- signs gives you the additive inverse which is equal at zero. However this property can be also be ...
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### Calculating Coefficents of a single variable polynomial [duplicate]

Given: $$(1+x+x^2+x^3+\cdots+x^k)^n$$ Is there a formula to calculate the coefficient of $x^a$ (where $a$ can be any integer value less than $k^n$) that's more efficient than grinding through ...
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### Algebraic forms for nth Roots of unity

Looking for algebraic forms for primitive nth roots of unity when n is composite. When n is prime it is simply all the possible roots of $\sum_{m=0}^{\phi(n)} X^M =0$ It's the composite values ...
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### A Dilogarithmic Sum

I have been playing around with dilogarithm and considered the following sum. $$\sum_{n=1}^{k} {\text{Li}_{2} {(\omega^n)}}$$ Where $\text{Li}_2{(x)}$ is the Dilogarithm and $\omega$ is the k-th root ...
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### What will be the number of distinct element in the set of the roots of unity?

Let $\omega$ denote a cube root of unity which is not equal to $1$. Then what is the number of distinct elements in the set {${(1+\omega+\omega^{2}+\cdots+\omega^{n})^{m}:m,n=1,2,3\cdots}$} My answer ...
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### $2$ as order $n$ unitroot in $\mathrm{GF}(2^n \pm 1)$: what should I know?

I'm currently trying to implement the DFT in Finite Fields in Schönhage Strassen matter as theoretical refreshment for the NTT (FFT variant of the NTT). My problem is that if I have an array of lets ...
I am studying a fake proof of FLT which assumes $\mathbb Q\left[\zeta_p \right]$ is a UFD (that is, that the rationals with the pth root of unity is a UFD). We have covered the proofs for n=2, 3, and ...