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

662 questions
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$\bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

Is $$\bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$$my argument is that the argument of the elements of the first set are rational multiples of $\pi$ whereas the second set also ...
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$\zeta_n^k$ is primitive if and only if $(k,n) = 1$ [duplicate]

Show that for $k \in \mathbb{Z}$: Is $\zeta_n$ a primitive $n$-th root of unity, then $\zeta_n^k$ is primitive if and only if $(k,n) = 1$. I only need the backwards direction: $\zeta_n^k$ is ...
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What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p$ contains all $(p-1)^{th}$ root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we ...
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How to get common roots of unity of $z^{6}=1$ and $z^{21}=1$?

How to get common roots of unity of $z^{21}=1$ and $z^{6}=1$ for $z\in\mathbb C$? I know that $z^{n} =1$ has roots $z=e^{\frac{2\pi k }{n}i}$ where $k\in \{0,1,2,...,n-1\}$
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Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
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Determine a Normal Basis for Galois Extension of $\mathbb{Q}$ with primitive pth root of unit (p prime)

Let p be a prime, $\xi_p \in \mathbb{C}$ a primitive p-th unit root and $K = \mathbb{Q}(\xi_p)$. Give a normal basis for $K/\mathbb{Q}$. I know, that a basis of $L/K$ (finite and galois) is ...
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If $y^4 = 5$ and $z^6 = 15$, then $y \notin \mathbb{Q}(z)$

Let $y,z \in \mathbb{C}$ with $y^4 = 5$ and $z^6 = 15$. I want to show that $y \notin \mathbb{Q}(z)$. So we have $$y = w_1 \cdot \sqrt5 \;\;\;\;\;\;\; z = w_2 \cdot \sqrt{15}$$ with $w_1$ ...
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What is the sum of the squares of the 10th roots of unity?

Obviously the sum of the roots of unity is 0, but is there a way to calculate this other than calculating them all individually and squaring them?
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Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

For each positive integer $n$, let's define the polynomial $$P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$$ Do the zeroes of $P_n$ lie inside, outside, or on the unit circle $|z|=1$? I tried ...
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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 ...
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cardinality of Galois group in $\mathbb{Q}$($\zeta_n$) [duplicate]

Let $\mathbb{Q}$($\zeta_n$) be some cyclotomic field, where $\zeta_n$ is a n-th root of unity. I already managed to show that $\mathbb{Q}$($\zeta_n$) is an Galois extension, but now i struggle to ...
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Showing the lemma $\operatorname{ord}_p(1+ζ_p)=0$ if $p>2$

Just to give some background regarding my motivation, I'm trying to prove a lemma to help me solve How do we prove p-order of $g_k$ is $\frac {k} {p-1}$? Let $Z_p$ denote the p-adic integers, and let ...
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Proving $ord_p(ζ_p-1)=1/(p-1)$

After proving this, I was able to deduce an even more general result that $ord_p(ζ_p-1)=ord_p(ζ_p^2-1)=...=ord_p(ζ_p^{p-1}-1)$. Now, according to Lubin, $ord_p(ζ_p-1)$ should be $1/(p-1)$, but this ...
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Proof that there are exactly $n$ distinct $n$th roots of unity in fields of characteristic zero?

I think it's true that in a field $F$ of characteristic zero, there are exactly $n$ distinct $n$th roots of unity (in some algebraic closure $\bar{F}$), that is, roots of the polynomial $x^n-1$. I ...
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Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book and it says that for $A =$ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
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Complex Endomorphism $f$ with orthogonal Eigenspaces implies that $f$ is unitary

I already posted this question earlier today without much success. After 3 hours and only 10 views I decided to give it another shot :) Of course I'll delete the second post! Let $V \ne 0$ be an ...
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Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$

i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$. ...
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How can I find the fifth root of unity?

I need to find fifth root of unity in the form $x+iy$. I'm new to this topic and would appreciate a detailed "dummies guide to..." explanation! I understand the formula, whereby for this question I ...
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holomorphic functions, roots of unity and harmonic numbers

If $f$ is a non-constant holomorphic function such that, for all $z \in \mathbb{C}$, exists a $c \in \mathbb{C}$ where $f(cz) = f(z),$ then $c$ must be a $n$-th root of unity, or there exists some ...
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Expressing $\zeta^k+\zeta^{-k}$ as a polynomial in $\zeta+\zeta^{-1}$.

Let $\zeta$ be an $n$-th root of unity and let $\chi:=\zeta+\zeta^{-1}$. Then $\zeta^k+\zeta^{-k}=P_k(\chi)$ where $P_k\in\Bbb{Z}[X]$ is a polynomial not depending on $n$. For example we have \begin{...
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Decomposition field of $x^n - 1$ over $Z_p$

In my textbook, it is said that (Primitive root) Let $p$ be a prime and $n > 1$ be a natural number. The set of all the roots $\alpha$ of the polynomial $x^n - 1 \in Z_p[x]$ forms a cyclic group ...
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if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then…

if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then $$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$ ...
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Number of roots of unity on the unit circle

I have to find the number of roots of unity on the unit circle |z|=1 in the argand plane. I know that there are n roots of the the equation $z^n=1$ and all of them lie on the given circle. Does that ...
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$\zeta$ is a p-th primitive untary root iff $-\zeta$ if a 2p-th primitive unitary root, with $p$ an odd prime

$\Rightarrow$ $(-\zeta)^{2p}=(-\zeta^p)^{2}=1$ and if $i \in[{1,2p-1}]$ there is $k \in[0,p-1]$ such that $i=2k+1$ if is odd and $2k$ if is even so $(-\zeta)^{i}=(-\zeta)^{2k}=(\zeta^k)^{2}\neq 1$ ...
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Prove that sum of n-th degree roots of complex number is 0

I'm trying to prove, that sum of all complex roots of n-th degree of a complex number $z$ is equal to 0. I know how to prove it for $z = 1$ (roots of unity), however i have to prove it for any complex ...
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I just cannot seem to wrap my head around this problem and would really appreciate some guidance. So I know that a primitive $n^{th}$ root of unity is a complex number $z$ such that $z^n = 1$ but $z^m ... 1answer 55 views Variance of sum of random roots of unity Let$X_{m,n}$be the sum of$n$uniformly random, independent samples from the set of all$m$th roots of unity. Obviously, the expectation of$X_{m,n}$is$0$. How do I go about reasoning about the ... 2answers 198 views Roots of$z^3 + 3iz^2 + 3z + i = 0$? I am asked to solve the equation in the title by first solving$\Big(\frac{z+1}{z-1}\Big)^n = i$, which is fine, but I can’t seem to manipulate this equation to get the equation. I’ve tried$n =$... 1answer 31 views Existence of full set of$k$-th roots of unity in$GF(p)$We all know that if$p$is prime then for$k = p-1$in$GF(p)$(field of integers mod$p$) all non-zero elements of the field constitute the full set of$k$-th roots of unity (Fermat's Little Theorem).... 1answer 146 views Primitive roots of unity and$I$-adically separated rings. Let$R$be an integral domain with$\operatorname{char}(R) = 0$and let$\zeta, \zeta'$be two primitive roots of unity in$R$. The following are equivalent. (1)$(q-\zeta)^m \in (q-\zeta') + I[q]$... 3answers 1k views Why isnt product of nth roots of unity always 1 I know product of nth roots of unity is 1 or -1 depending whether n is odd or even. But in this way I am getting 1. Where am I wrong?$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \...
Consider the combinatorial identity by Gould, Table III, page 25, equation (6.13): \sum_{k=0}^{[\frac{n}{r}]}{n \choose rk}=\frac{2^n}{r}\sum_{j=1}^{r}\left(\cos{\frac{\pi j}{r}}\right)^n\cos{\frac{...