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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

An $n$-th root of unity is a complex number $z$ such that $z^n=1$ for some $n\in \mathbb N$. If $n$ cannot be replaced by a smaller natural number, then $z$ is called primitive $n$-th root of unity. There are $\varphi(n)$ primitive $n$-th roots of unity and they are roots of the $n$-th cyclotomic polynomial (which has degree $\varphi(n)$). The $n$-th roots of unity can be written as $e^{ \frac{2k\pi}n\cdot i}$ with $0\le k\lt n$.

An important lemma: if $z$ is an $n$-th root of unity, $$ \sum _{k=0}^{n-1} z^k = \begin{cases} n,& z=1 \\ 0,& z\neq 1\end{cases} $$In particular if $z$ is a primitive $n$-th root the sum is zero, a property commonly used in elementary number theory.

The concept can be extended to other fields than $\mathbb C$. For example, in a finite field with $q$ elements, all non-zero elements are $(q-1)$-th roots of unity.

See also this Wikipedia article.

1062 questions
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Complex roots of unity arrangement.

The following question is from an entrance test named IAT (Indian Institute of Science and Education Research Aptitude test). Let $1, z_2, z_3,\ldots, z_n$ be the roots of the equation: $$x^n-1=0,\quad n\ge3$$ Find the value of:…
Kinjal
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For $1 \not= \alpha \in \mathbb{C}$ such that $\alpha^7 = 1$, evaluate $\alpha + \alpha^2 + \alpha^4.$

For $1 \not= \alpha \in \mathbb{C}$ such that $\alpha^7 = 1$, evaluate $\alpha + \alpha^2 + \alpha^4.$ My solution : let $$p = \alpha + \alpha^2 + \alpha^4$$ and $$q = \alpha^3 + \alpha^5 + \alpha^6.$$ We know $$1 + \alpha + \alpha^2 + \alpha^3 +…
Vue
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"Converse" to the theorem "sum of roots of unity equal 0"

It is well known that sum of roots of unity equal 0. However, if $\sum_j \exp(i \phi_j)=0$, can we say something about the relation between the $\exp(i \phi_j)$'s? For example, we can rotate one of them to the position of 1, can we say that the…
enochk.
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Show that $\frac{3}{5} + i\frac{4}{5}$ isn't a root of unity

Intuitively, I see why $\frac{3}{5} + i\frac{4}{5}$ is not a root of unity because $\frac{2\pi}{\arctan(4/3)}$ appears to be irrational when I plug into my calculator. But how to I show this rigorously?I think contradiction should work, but still I…
Debbie
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Identity regarding roots of unity

Let $\zeta$ be a primitive $n$-th root of unity and $m \in \{0,1,\dots,n-1\}$. I am interested in finding the value of the following expression: $$\sum_{k=1}^{n-1}\frac{\zeta^{mk}}{1-\zeta^k}.$$ This has come up in a context where it should be a…
Miguel González
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Do the $n$-th roots of unity of an *arbitrary* field form a cyclic group?

Do the $n$-th roots of unity of an arbitrary field form a cyclic group? Or stated differently, can we always find a primitive $n$-th root of unity? Because if we have this element we can generate the group $<\zeta_n>$, and we are done. In…
Jens Wagemaker
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Sum of a proper subset of the $p^\text{th}$ roots of unity

I know, because I have read it, that if $p$ is prime, no sum of a proper subset of the $p^{\text{th}}$ roots of unity (in $\mathbb{C}$) is zero. I thought I knew how to prove this, but found to my dismay when I tried that I do not. Can someone give…
Leon Avery
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Textbook says roots of unity is equal to 1

The elements of the set $U_n = \{z \in \mathbb{C} : Z^n =1 \}$ are called the $n^{\text{th}}$ roots of unity. Using the technique of Examples 1.6 and 1.7, we see that the elements of this set are the numbers $$\cos \left(m \frac{2\pi}n \right)+i…
Åge83
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Finding the qth root of unity mod p

I have $p$ and $q$ as $p = 4916335901, q = 88903$ and I have to find the $q^{th}$ root of unity $\pmod{p} $, so its $q{th}$ root of unity $\pmod{4916335901}$. What exactly is a $q^{th}$ root and what would it be for this? I have used this wiki page…
orange
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Minimize sum of $k$ distinct $n$-th roots of unity.

Let $0
AndreA
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Find $\sum_{k=1}^n P(z_k)$ where P is a polynomial and $z_k$ the $k$th root of $z^{13}=1$

Question: Given a complex number z such that $z^{13}=1$, find the sum of all possible values of $z+z^3+z^4+z^9+z^{10}+z^{12}$. I know we have to use roots of unity and try to manipulate the polynomial we want to evaluate, but can't find a pattern.
user829751
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If $1,a_1,a_2,...,a_{n-1}$ are the $n$ roots of $1$,then $(1-a_1)(1-a_2)...(1-a_{n-1}) \space ?$

If $1,a_1,a_2,...,a_{n-1}$ are the $n$ roots of unity, then how can we find the value of $$(1-a_1)(1-a_2)...(1-a_{n-1}) \space ?$$ My Approach: If $n=2$ , then $1,-1$ are the roots of unity $\therefore (1-a_1)=(1-(-1))=2$ for $n=3 \space :$ …
Suresh
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Most efficient way of calculating primitive cube roots of unity

I understand the definition of a primitive cube root of unity in a finite field $\mathbb{F}_p$ to be all those numbers $x$ such that $x^3=1$ but $x\neq 1$ and $x^2 \neq 1$ When we have a small $p$, say $p=7$, we can compute these through 'brute…
lioness99a
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Problem involving roots of unity

Let $\varepsilon _k$ be $\cos \frac {2k \pi} {n} + i \sin \frac {2k \pi} {n}$. Find the value of the product: $$\prod _ {k=1}^n (2+\varepsilon _k-\varepsilon _k^2).$$
M. Stefan
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Find roots of unity

Find the roots of $6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 = 0.$ I know how to do this without the coefficients, but I do not know what to do in this problem. Thanks
Rohini Sharma
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