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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.

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Primitive roots of unity in $\mathbb{Z}/p$

Can anyone help me with this problem? Let $p$ be a prime number. Prove that if the field $\mathbb{Z}/p$ has a primitive $n^{th}$ root of unity, then $n \mid (p-1).$ Any sources or books for reference? Any hints also appreciated.
Strider2
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$\forall n \in\mathbb N$, determine which $n$-th root of unity is closest to $\frac12$.

$\forall n \in\mathbb N$, determine which $n$-th root of unity is closest to $\frac12$. I'm really struggling with where to even begin with this question. Any help would be appreciated
idkkkkkkk
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7th roots of unity

I'm having difficulty expressing 7th roots of unity with nth root. Here in the photo are my solutions, but they are too complicated and I'm not sure if they are correct. If you would tell me if they are correct, or how to solve the problem, that…
sally
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Solving roots of unity

The three cube roots of unity are $\omega$, $\omega^2$ and 1, where $$\omega = \frac{-1+\iota\sqrt{3}}{2}, \qquad\omega^2 = \frac{-1-\iota\sqrt{3}}{2}$$ Evaluating the equation $$x^3 - y^3 = (x-y)(x-\omega y)(x - \omega^2y)$$ My attempt : Solving…
Muhammad Salman
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Nth roots of unity

Find two roots of unity such that their sum equal 1 Okay so what I did, is that I said that: $$\exp((2i\pi)/n)+\exp((2i\pi)/n)=1$$ which was equivalent to $$2\exp((2i\pi)/n)=1$$ $$\exp((2i\pi)/n)=\frac 1 2$$ But then only non-sense came from…
Student number x
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n-th primitive root of unity

Let $p>3$. Suppose $f$ is third primitive root of unity in $F_p$. I need to prove the following: $(2f+1)^2=(-3)$ . So, what I did: $f^3=1$ but it doesn't hold for first and second power of $f$. Then, $(2f+1)^2 = 4(f^2 + f + 1) - 3$ But then, if I…
Ekber
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Cyclotomic Polynomials

Let $E(n)$ denote an nth root of unity. (For convenience, we may take $E(n) = \exp(\frac{2πi}{n})$.) Prove that for any prime $p$ and any natural number $r$, we have $$ \prod_{\substack{j\\ \gcd(p^r, j) = 1}} \left(1 – E(p^r)^j\right) = p,$$…
John Chang
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Why $a_o + a_1 \omega ... + a_{p-1} \omega^{p-1} = 0$ implies that $a_0 = a_1 = ... = a_{p-1}.$

Suppose $w$ is $p$-th root of unity. Why when from $$a_o + a_1 \omega + a_2 \omega^2 ... + a_{p-1} \omega^{p-1} = 0$$ we can conclude that $$a_0 = a_1 = a_2 = ... = a_{p-1}.$$ And later use it to find some nonzero $a_i.$ I have found this piece in…
Yola
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A theorem involving the special root of an equation

Theorem: If $a$ is a special root of the equation $x^{n}-1=0$, then $a^{p}$ is also a special root of it (where $p$ is prime to $n$). I have done a proof of this theorem. Can you please tell if my method is incorrect? My Method: as $a$ is a…
Avijit Mukherjee
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Sum of roots of unity with linear coefficients

Given that $n$ is a positive integer and $\omega=e^{2\pi i/n}$, how do you prove that $a_0+a_1\omega+a_2\omega^2+...+a_{n-1}\omega^{n-1}=0$ is only true when all the $a$'s are equal?
Hanson Bai
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Quadratic Gauss Expression for primitive $7$th roots of unity

For primitive $7$th root of unity $\omega$, calculate $|1+2\omega + 2\omega^2 + 2\omega^4|$.
Vivek A
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Finding the elements of a number C based off of 24th and 54th roots of unity

Let $A$ be a set of all complex numbers $z$ such that $z^24=1$ and let $B$ be the set of all complex numbers $w$ such that $w^54=1.$ That is: \begin{align*}A&=\{z\;|\;z^{24}=1\}\\ B&=\{z\;|\;z^{54}=1\}\\ \end{align*} Finally, let $C$ be the set of…
majorbill
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nth root of unity problem for homework

I'm super confused right now with this problem and any kind of mini lecture would be awesome. Thank you Let $ω = e^{2πi/n}$ , where $n$ is a positive integer. Prove that (a) $1 · ω · ω^2 · · · ω^{n−1} = (−1)^{n−1}$ .
Tony Mau
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