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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

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1answer
36 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 $w_1$ ...
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3answers
41 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
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1answer
252 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
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1answer
74 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 ...
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1answer
50 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 ...
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0answers
25 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\...
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2answers
97 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 ...
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1answer
34 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 ...
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3answers
32 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 $n$. \...
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1answer
24 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 ...
1
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1answer
48 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 ...
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3answers
43 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 ...
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0answers
35 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 ...
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0answers
16 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 ...
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0answers
36 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 $\mathbb{Q}[x]$. ...
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2answers
39 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 ...
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1answer
39 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{...
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0answers
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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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1answer
43 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 $\...
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3answers
109 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
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1answer
51 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 ...
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1answer
15 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 $(-\zeta)^{i}=(-\zeta)^{2k}=(\zeta^k)^{2}\neq 1$ ...
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3answers
50 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 ...
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1answer
228 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 ...
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0answers
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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 $p$-...
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2answers
42 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 ...
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22 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}} \...
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0answers
38 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 ...
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1answer
46 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 ...
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1answer
26 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)....
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1answer
46 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 ...
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0answers
187 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{...
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1answer
36 views

$n$-th root of unity implies $d$-th primitive root of unity for some $d\big|_n$

Let $F$ be a field and let $n \in \mathbb{N}$ and let $\varepsilon \in \overline{F}$ ($\overline{F}$ is an algebraic clausure of $F$) be a $n$-th root of unity ($\varepsilon$ is a root of $X^n - 1 \in ...
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2answers
57 views

Roots of unity and large expression

Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \...
0
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1answer
27 views

If $\zeta_n$ is a primitive $n$th root of unity, why is $\text{dim}_{\Bbb Q}\Bbb Q[\zeta_n]=\phi(n)$? [duplicate]

I have no idea what cyclotomic polynomials are and how we can get the result using that. Is there another way to prove it? Any hint is appreciated.
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1answer
38 views

If $\zeta_n$ is the $nth$ root of unity, find $\dim_{\Bbb Q}\Bbb Q[\zeta_n]$

For $n=1,2,3,4$ I found it to be $1,1,2,2$ respectively. So I hypothesize that $\dim_{\Bbb Q}\Bbb Q[\zeta_{2n}]=n$ and $\dim_{\Bbb Q}\Bbb Q[\zeta_{2n+1}]=n+1$. But how to prove it? When we have $2n$, ...
2
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1answer
51 views

For what integers $n$ is $-1$ an $n$th root of unity?

Can someone please verify my answer? I feel like I made it too complicated or I am missing something. For what integers $n$ is $-1$ an $n$th root of unity? There are two values on the unit circle ...
0
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1answer
34 views

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)...
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0answers
40 views

Summation involving Roots of Unity

Problem Let $x \in \mathbb{R_{\geq 0}}$, $\alpha \in (0,1]$ and $p \in \mathbb{N}$. Define $$S(x)=\sum_{k=0}^{p-1} \frac{(e^{\frac{2\pi k}{p}i})^{x+1}}{e^{\frac{2\pi k}{p}i}-\alpha} $$ Does there ...
1
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1answer
26 views

An Upper and Lower Bound for a Field Extension of $\mathbb{Q}$ by a Complex Root

Consider the polynomial $f(x) = x^3 + \zeta x + \sqrt{3}$ in $\mathbb{C}[x],$ where $\zeta$ is a primitive third root of unity. Given a root $\alpha$ of $f(x)$ in $\mathbb{C},$ prove that $4 \leq [\...
3
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1answer
53 views

Finding roots of unity

Exercise Let $w, z$ are the unit roots of equation $z^5 = 1$ for $z \in \mathbb{C}$. Prove that $(w^{21} + z^{14})^5$ is always a real number. So, first I can do: $w^{21} = w$ $z^{14} = z^4$ So ...
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0answers
62 views

Closed formula for $\Phi'_n(\zeta)$ as complex number

This is a follow-up to this question. The one thing that did not get a completely satisfactory answer there is: If $\Phi_n$ denotes the $n$-th cyclotomic polynomial, and $\zeta^k_n = e^{2k\pi i/n}$ is ...
3
votes
1answer
145 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) $(q-\zeta)^m \in (q-\zeta') + I[q]$ ...
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2answers
51 views

An interesting property of the roots [closed]

Let $x^2+x+1=0$ be a quadratic equation having two roots $x_1$ and $x_2$. How would you prove that $(x_1)^2=x_2$ and $(x_2)^2=x_1$? P.S. I could show $(x_1)^2=x_2$ and $(x_2)^2=x_1$ by finding the ...
4
votes
4answers
148 views

Complex Partial Fraction Decomposition

The question I need help with is: Prove that $$\sum_{k=0}^{6}\frac{1-z^{2}}{1-2z\cos\left(\frac{2k\pi}{7}\right)+z^{2}}=\frac{7(z^{7}+1)}{1-z^{7}}$$ I have already tried brute forcing this by ...
0
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0answers
49 views

Critical coefficient for recurrence relation and characteristic polynomial.

I am simulating the behavior of a system I have where I apply feedback based on the error I estimate with an averaging window of length $M$, and some gain $g$. I have come up with the following ...
0
votes
1answer
80 views

Root of unity belongs to Z/qZ. How?

EDIT: Really sorry for not posting this initially.. maybe it's easier to understand now. Source, page 6. I've stubled upon a statement similar to this: "Let $m,q$ be two integers such that $\mathbb{...
0
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1answer
47 views

The system $2XY+X^2+Y^2+X^2Y+Y+XY^2+X=0$, for distinct pairs $X, Y\in\{J, K, L\}$ for $n$th roots of unity $J, K, L$.

Solve the system $(I)$ of equations $$\begin{align} 2JK+J^2+K^2+J^2K+K+JK^2+J &=0,\tag{$Ia$} \\ 2KL+K^2+L^2+K^2L+L+KL^2+K &=0,\tag{$Ib$} \\ 2LJ+L^2+J^2+L^2J+J+LJ^2+L &=0\tag{$Ic$} \end{...
2
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0answers
86 views

Primitive $p$-th root of unity with characteristic $p$

I struggle on this since two days, and still found no answer. My course states the following: If the characteristic of $K \neq p$, then they are exactly $p-1$ different $p$-th roots of unity in the ...
4
votes
3answers
245 views

What exactly is an $n$-th root in a field?

What is meant when one says that a finite field contains an $n$th root of $a$? Is it an exquisite way to say that the multiplicative group of the field has an element $x$ such that $x^n=a$? If $a=-1$, ...