# 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

647 questions
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### Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
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### What online graphing tools handle complex numbers well?

What online graphing tools handle complex numbers well? Desmos is generally excellent by breaking functions down into their real and imaginary parts and plotting on the Euclidean plane. For example ...
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### Question about roots of unity in the Fast Fourier Transform

I am learning about the Fast Fourier Transform, which converts a polynomial from its coefficient representation into its point-wise form using divide-and-conquer. The Fast Fourier Transform evaluates ...
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### Roots of unity divisibility.

Suppose $r | n$. Then $R:= e^{2i \pi k/r}$ is an $n$-th root of unity. Thus, there exists a unique $l \in \{0, \dots, n-1\}$ such that $R = e^{2\pi i l/n}$. Does it hold that $l |n$? I tried to ...
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### If $a \in \mathbb{C}$ and $\exists n \in \mathbb{N}$ s.t. $\{ a^n, a^{n+1} \} \in \mathbb{N}$, prove $a \in \mathbb{N}$ [closed]

Let $a$ be a complex number. If it exists a natural number $n$ (different of $0$), such that $a^n$ and $a^{n+1}$ are integers, prove that $a$ is an integer.
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### Do there exist non-trivial integer coefficients that break linear independence of the roots of unity?

Let $n$ be a positive integer, $k=0,\cdots,n-1$, $\omega_k=e^{\tfrac{2\pi i}{n}k}$ be the roots of unity, $c_k \in \mathcal{Z}$ be integer coefficients, trivial and non-trivial be two subcategories ...
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### To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$

Sum the series: $$\frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!},$$ $n$ being a multiple of $3$.(Math. Tripos, 1899) My attempt We may ...
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### If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

A lemma in my lecture notes states that if $K$ is a subfield of $\mathbb{C}$ and $\zeta=\exp(2\pi i/p)$ then $K(\zeta)/K$ is Galois. They proved it by arguing that the minimal polynomial of $\zeta$ ...
### If $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$.
How do I prove that for all natural numbers $n$ and complex numbers $a, b, c, z, w$ if $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$. ...