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

6
votes
2answers
675 views

$ \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 ...
0
votes
1answer
48 views

$\zeta_n^k$ 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 ...
7
votes
1answer
297 views

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 ...
1
vote
3answers
120 views

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\} $
4
votes
2answers
105 views

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 ...
4
votes
1answer
98 views

Why is the magnitude of the sum of two adjacent nth roots always an 'interesting' number, and what do these numbers have to do with each other?

While doing something completely unrelated, I discovered an interesting function: $$f(x)=2\left\vert\cos{\frac{\pi}{x}}\right\vert$$ Which gives the absolute value of the sum of any two adjacent $x^\...
2
votes
1answer
58 views

Finding the Normal Basis of Cyclotomic field

So let $p$ be a prime number and $\zeta_p$ the p-th roots of unity. I want to proof that $ B = \{ \zeta_p, \zeta_{p}^{2}, \dots, \zeta_{p}^{p-1} \}$ is the normal basis of $\mathbb{Q}(\zeta_p)/\mathbb{...
0
votes
0answers
59 views

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 ...
2
votes
1answer
62 views

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 $$ y = w_1 \cdot \sqrt[4]5 \;\;\;\;\;\;\; z = w_2 \cdot \sqrt[6]{15} $$ with $w_1$ ...
0
votes
3answers
72 views

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?
7
votes
1answer
368 views

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 ...
6
votes
1answer
77 views

“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 ...
0
votes
1answer
76 views

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 ...
1
vote
0answers
33 views

5-adic order of $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$

Let $\pi:$(the 5-adic integers)$\to \mathbb{Z}/5\mathbb{Z}$ be the reduction map. Let $f:\mathbb{Z}/5 \mathbb{Z} \to $ (the 5-adic integers) have the following properties $\forall x,y\in \mathbb{Z}/5\...
4
votes
2answers
118 views

Compute $\sum\frac1{2-A_k}$ for $(A_k)$ the $n$th roots of unity [duplicate]

If $1,A_1,A_2,A_3....A_{n-1}$ are the $n^{th}$ roots of unity then prove that $$\dfrac{1}{2-A_1} + \dfrac{1}{2-A_2}+\cdots+ \dfrac{1}{2-A_{n-1}} = \dfrac{2^{n-1}(n-2) + 1}{2^n-1}$$ What I did: I ...
0
votes
1answer
27 views

Roots of sparse “quadratic-like” polynomial.

So I know about this question and I've seen papers like this and this. But the former isn't exactly what I want and the latter two papers are too deep and I'm lazy and I wanna quick-and-easy answer ...
2
votes
1answer
51 views

Are the prime cyclotomic polynomials irreducible over any field where they're not obviously reducible ?

My question is the following : if $p$ is a prime number, $\Phi_p = \frac{X^p-1}{X-1}$, is $\Phi_p$ irreducible over any field $K$ where it has no root ? Phrased differently, if $K$ is of ...
1
vote
3answers
36 views

Relation between the numbers of units, square roots of unity and divisors for the rings $\mathbb{Z}/n\mathbb{Z}$

For the non-prime numbers $n$ up to $20$ I listed the number of units of $\mathbb{Z}/n\mathbb{Z}$, the number of square roots of unity of $\mathbb{Z}/n\mathbb{Z}$ and the number of divisors of $n$. \...
8
votes
1answer
236 views

Find $P(x,y,z)=x^n+y^n+z^n-\prod\limits_{k=0}^{n-1}(x+\omega_n^ky+\omega_n^{-k}z)$, where $\omega_n$ denotes a primitive $n$th root of unity

Find $P(x,y,z)=x^n+y^n+z^n-\prod\limits_{k=0}^{n-1}(x+\omega_n^ky+\omega_n^{-k}z)$, where $\omega_n$ denotes a primitive $n$th root of unity. I have manually multiplied the terms of the product and ...
1
vote
1answer
43 views

Find a polynomial in $\mathbb{Z}_{41}$

Find a $7^\text{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}[x]$, so that $$ p(14^i) = i\pmod{41}\ \forall i = 0,1,\ldots,7. $$ Hint: $3$ is the $8^\text{th}$ primitive root of unity and $3 \...
2
votes
3answers
47 views

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 ...
1
vote
1answer
55 views

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 ...
1
vote
1answer
76 views

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 ...
0
votes
0answers
96 views

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 ...
1
vote
0answers
17 views

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 ...
0
votes
0answers
40 views

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 $\mathbb{Q}[x]$. ...
4
votes
2answers
18k views

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 ...
3
votes
2answers
45 views

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 ...
1
vote
1answer
62 views

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{...
2
votes
0answers
74 views

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 ...
7
votes
2answers
208 views

Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.

I am trying to solve the following exercise in Koblitz's "$p$-adic Numbers, $p$-adic analysis, and Zeta-Functions". Let $p$ be a prime. Let $a$ be a primitive $p^n$-th root of unity in $\bar{\mathbb{...
3
votes
1answer
977 views

About the Center of the Special Linear Group $SL(n,F)$

I want to classify the Center of the Special Linear Group. I already determined the center for SL(n,F) its: $Z(SL(n,F))=\left\{ \lambda { I }_{ n }:\quad { \lambda }^{ n }=1 \right\} $ I showed ...
1
vote
1answer
51 views

If $\zeta$ is an $m$th root of unity, then $1 - \zeta^k \in \mathfrak{q}$ implies $1 -\zeta^k = 0$

Let $m \in \mathbb{Z}$ such that $m$ is not a prime power, and suppose that $\zeta$ is a primitive $m$th root of unity. Let $q$ be a prime number such that $q$ doesn't divide $m$, and suppose that $\...
5
votes
3answers
179 views

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 $$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$ ...
1
vote
1answer
168 views

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 ...
0
votes
1answer
18 views

$\zeta$ 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 $(-\zeta)^{i}=(-\zeta)^{2k}=(\zeta^k)^{2}\neq 1$ ...
0
votes
3answers
197 views

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 ...
1
vote
3answers
87 views

If $n>3$ prove that $\sum_{k=0}^{n-1} (k-n)\cos\frac{2k\pi}{n}=\frac{n}{2}$.

Do you have any ideas on this IIT exercise? If $n>3$ is an integer, prove that $$\sum_{k=0}^{n-1} (k-n)\cos(2kπ /n) = n/2$$ In my attempt, I have considered $$z=cis(2kπ/n), k=[1, 2,..., n-1]...
0
votes
0answers
19 views

About roots of unity in an arbitrary CM field K with an abelian extension L

I have an arbitrary CM field $K$ and an abelian extension $L$ of $K$ with dimension $[L:K]=p$, with $p$ prim. Suppose there exist a non trivial primitiv root of unity $\xi$ in K. Must $\xi$ be a $p$-...
2
votes
3answers
84 views

Let $w$ be a primitive root of a unit of order 3, prove that $(1-w+w^2)(1+w-w^2)=4$

The title is the statement of the problem. I did the following: $(1-w+w^2)(1+w-w^2)=$ $1+w-w^2-w-w^2+w^3+w^2+w^3-w^4=$ $1-w^2+w^3+w^3-w^4=$ $1-w^2+1+1-w^4=4$, * then,by definition of primitive ...
1
vote
2answers
53 views

Rings with a finite set of units

Consider a ring $\mathcal{R}$ with a finite set $\mathcal{R} ^\times$ of units, i.e. divisors of $1$, for example $\mathbb{Z}^\times = \{\pm 1\}$ $\mathbb{Z}[i]^\times = \{\pm 1,\pm i\}$ (Gaussian ...
26
votes
7answers
14k views

Intuitive understanding of why the sum of nth roots of unity is $0$

Wikipedia says that it is intuitively obvious that the sum of $n$th roots of unity is $0$. To me it seems more obvious when considering the fact that $\displaystyle 1+x+x^2+...+x^{n-1}=\frac{x^n-1}{x-...
0
votes
0answers
24 views

Quadratic Gauss sum using $|Z|^2=Z \bar Z$

Given $Z=\sum_{k=1}^{n-1}\omega^{k^2}$ I'm asked to find $|Z|^2$, here's what I thought of: $$|Z|^2=Z \bar Z=\left( \sum_{k=1}^{n-1}\omega^{k^2} \right) \left( \sum_{k=1}^{n-1}\frac{1}{\omega^{k^2}} \...
1
vote
0answers
145 views

Primitive nth roots of unity related to the complex nth roots of 1.

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 ...
0
votes
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 ...
0
votes
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 $n =$ ...
1
vote
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)....
3
votes
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) $(q-\zeta)^m \in (q-\zeta') + I[q]$ ...
2
votes
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 \...
7
votes
0answers
203 views

Sum of binomial coefficients in Gould tables

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{...