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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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Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

Let $c_k(n)$ denote Ramanujan's sum, and $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Prove that $$\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$$ My attempt was to ...
Mako's user avatar
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$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
muhammed gunes's user avatar
3 votes
0 answers
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Natural map of automorphism groups

Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
ByteBlitzer's user avatar
1 vote
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Resources on infinite series involving $n$'th roots of unity

Background I was wondering whether there any books, articles, or other in-depth treatments of infinite series involving the roots of unity. Let $\omega_{n} := e^{2 \pi i / n}$ be the $n$'th root of ...
Max Muller's user avatar
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1 vote
1 answer
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Help understanding derivation of identity $\sum_{k=1}^{n}\cot^4\left({k\pi\over 2n+1}\right)=\frac{1}{45}n(2n-1)(4n^2+10n-9)$

This question regards understanding some of the steps in the derivation of the identity for $\sum_{k=1}^{n}\cot^4\left({k\pi\over 2n+1}\right)$ It is shown 1 that using Vieta's formula that $\sum_{k=1}...
onepound's user avatar
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3 answers
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Can we find the root of this equation

Give an equation below: $$ \frac{x^k-a^k}{x-a}=c \qquad (1) $$ where $1<a<x$, $0<k<1$, and $c>0$. I can easily find the numerical root of (1) by using Newton's method or the other tools....
Tyke's user avatar
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3 votes
2 answers
212 views

Generalization of Integer-Powered Sums Problem

I am trying to solve a problem that involves the sum of the $n$-th roots of positive reals. Specifically, the task is to determine all sets of positive reals $a_1, a_2, a_3$ such that $\sqrt[n]{a_1}+\...
Snowball's user avatar
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5 votes
1 answer
288 views

Understanding roots of unity in quadratic fields

Suppose we have a quadratic field $\mathbb{Q}(z)$ with $z \in \mathbb{C} \setminus \mathbb{Z}$ and $z^2 \in \mathbb{Z}$. How would one go about determining the possible $n^{\text{th}}$roots of unity ...
Ben1669's user avatar
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4 votes
1 answer
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Linear Dependence of Primitive Roots of Unity

Consider the cyclotomic field $\mathbb{Q}(\zeta_n)$. We know that the set of primitive roots $\Pi_n=\{\zeta_n^m:(m,n)=1\}$ generates $\mathbb{Q}(\zeta_n)$ as a field. However, what happens when we ...
wakewi's user avatar
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Certain sum involving roots of unity like the Lambert series

Let $\ell$ be an odd prime $, \zeta=e^{\frac{2 \pi i}{\ell}}, v \in \mathbb{Z}_{>0}$. Then how to prove there exist an integer $N \equiv -v \ (\bmod \ell)$, $$ \begin{aligned} \frac{\zeta^v}{(1-\...
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6 votes
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Anything interesting known about this generalization of even and odd functions?

Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$, $$f(\omega z) =...
Torsten Schoeneberg's user avatar
1 vote
1 answer
60 views

Degree of field extension $\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q$

$n>2$ is an odd squarefree integer. Let $\zeta_n=\mathrm{e}^{i\frac{2\pi}n}$ be a primitive $n$-th root of unity. Is it true that $[\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q]=\...
hbghlyj's user avatar
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6 votes
2 answers
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The unit digit of $\prod_{k=0}^{97}\left(2+\alpha_{k}^2\right)$, where $\alpha_0,\alpha_1,....,\alpha_{97}$ are the $98^{th}$ roots of unity

The unit digit of $$\prod_{k=0}^{97}\left(2+\alpha_{k}^2\right)$$, where $\alpha_0,\alpha_1,....,\alpha_{97}$ are the $98^{th}$ roots of unity My Approach: Since $\alpha_0,\alpha_1,....,\alpha_{97}$ ...
mathophile's user avatar
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Summation of reciprocals of nth roots of unity

I let the term inside the summation be equal to x, put $a$ in terms of x and used $a^n$=1 but because I have not done binomial yet I got stuck at that point. I was thinking of opening using binomial ...
Toshiv's user avatar
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2 votes
1 answer
105 views

Let the value of $\log_{2} \left[\prod_{a=1}^{2015} \prod_{b=1}^{2015} \left(1+e^{\frac{2 \pi i a b}{2015}}\right)\right]$

Let the value of $$\log_{2} \left[\prod_{a=1}^{2015} \prod_{b=1}^{2015} \left(1+e^{\tfrac{2 \pi i a b}{2015}}\right)\right]$$ is $N$, then which of the following is/are true (a) $N$ is divisible by $5$...
mathophile's user avatar
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2 votes
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The primitive $n^{th}$ roots of unity form basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity iff $n$ is square free

Prove that the primitive $n^{th}$ roots of unity form a basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity if and only if $n$ is square free I think I have the $(\Rightarrow)$...
Grigor Hakobyan's user avatar
-1 votes
1 answer
43 views

$z^p =(z^n)^{p/n} =1$? where z is $n^{th}$ root of unity and $p \in \mathbb R$ [duplicate]

We know that there are $n$ solutions for $n^{th}$ root of unity, $z^n = 1$ : $1, z, z^2, z^3, ......z^{n-1}$ where, $z = e^{i2\pi/n}$ Now, $$z^{n+1} = z$$ right? Since, $$z^{n+1} = z^n.z = 1.z=z $$$$.:...
Ishant Dumane's user avatar
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Field trace of complex roots of unity

I have been thinking about the field traces of roots of unity. Let $\zeta$ be a primitive $n$-th root of unity, and let $K$ be any subfield of $\mathbb{Q}[n]$. If I take the trace of $\zeta$ down to $...
Chris's user avatar
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0 votes
0 answers
41 views

In formula $x_k=\cos(2k \pi/n) +i \sin(2k \pi/n)$ why does $k$ goes from $0$ to $n-1$?

Formula is for finding roots of unity. I know to prove that it will work for any k,but can't see in the formula why k needs to be from $0...n-1$. Obviously equation $x^n -1=0$ has $n$ roots,but why ...
Stephanie V's user avatar
3 votes
1 answer
97 views

Particular sum of roots of unity

I've gotten stuck on a particular sum, to which I think I know the answer (thanks to Wolfram:Alpha), but not the method leading to it. I wonder if someone here can help me solve it. Let $d$ be a ...
ChangedMyName's user avatar
0 votes
0 answers
16 views

Bounds on randomized sums of roots of unity

Let $p$ be a large prime, and $S$ a set of non-negative integers less than $p$. Consider the distribution $D_{S,p}$ over complex numbers defined as $\sum_{x\in S}e^{i2\pi x y/p}$ for $y$ chosen ...
AAA's user avatar
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0 votes
1 answer
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Let $n$ be a prime number. Consider $\mathbb U_n$ which is the set of roots of unity. [duplicate]

Let $n$ be a prime number. Consider $\mathbb U_n$ which is the set of roots of unity, i.e. the solutions of the equation $z^n=1$, more precisely $\mathbb U_n = ${$1,\epsilon , \epsilon^2 , ... , \...
Unknowduck's user avatar
-1 votes
2 answers
105 views

About Group of nth roots of unity [closed]

Let $z \in \mathbb{C}$ and $n$ an integer $>3$. Define $U$ as the set of all $\lambda \in \mathbb{C}$ such that $\lambda^n = 1$. I aim to prove the following implication: $ (\forall \lambda \in U; (...
yassine's user avatar
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2 votes
0 answers
42 views

A question about how to represent $\mathbb{Q}(\zeta_n)$ differently

I am reading Dummit & Foote for studying undergraduate level field theory, and here is what I see on Page 555: I followed the words of writers and understood the proof for a statement $$\mathbb{Q}...
ZYX's user avatar
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1 vote
1 answer
96 views

When can non-trivial roots of unity be partitioned such that product of their sums is an integer?

We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$ This happens because ...
Trebor's user avatar
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0 votes
0 answers
60 views

Connected components of subgroup of torus

Consider a finite field $\mathbb{F}_q$ of characteristic $p>0$. Let $A=(a_{ij})$ be an integer matrix with $k$ columns and a finite number of rows. Consider the algebraic subgroup $\pmb{H}_A$ of ...
user avatar
4 votes
2 answers
88 views

Kummer's Lemma and $1+\zeta$

In lecture we were told to think about the following: Kummer's Lemma: Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ ...
3nondatur's user avatar
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3 votes
0 answers
152 views

How do I compute the eleventh roots of unity algebraically, expressed with only square roots and fifth roots?

I was able to reduce the problem to solving the quintic equation $$x^5+x^4-4x^3-3x^2+3x+1=0,$$ which seems to be the minimal polynomial of $2\cos(2\pi/11)=e^{2\pi i/11}+e^{-2\pi i/11}$. But I couldn't ...
Finn Bolton's user avatar
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0 answers
28 views

Interpolating polynomial for characteristic function of primitive Nth roots of unity among all Nth roots

In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose ...
user43208's user avatar
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3 votes
0 answers
89 views

On a step on a complex inequality with summation.

To make it short I have a doubt in the last step of a complex number inequality, the problem is the next one Let complex numbers $z_1,z_2,z_3,...,z_n$ all modulus $1$ and $z_1+z_2+z_3+...+z_{2012}=0$ ...
Ruben's user avatar
  • 127
1 vote
0 answers
73 views

Rational polynomial $f$ of minimal degree such that $f(x^{d-1},y^{d-1},z^{d-1})$ is divisible by $x^d+y^d+z^d$

$d\in\Bbb Z^{\ge2}$. What is the minimal degree of $f\in\mathbb{Q}[x,y,z]$ such that $f(x^{d-1},y^{d-1},z^{d-1})$ is divisible by $x^d+y^d+z^d$? Let $\zeta=\exp(\frac{2\pi i}{d-1})$. Since $f(x^{d-1},...
hbghlyj's user avatar
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2 votes
1 answer
143 views

Sum of complex numbers at the vertices of a regular polygon

If $z_1,z_2,z_3, \ldots ,z_n$ are the vertices of an $n$-sided regular polygon with $z_0$ as its centre, then find $$\sum_{r=1}^{n} z_r^k$$where $k \in \Bbb N, k<n$. And my working is here: $$z_r ...
The Techissta's user avatar
0 votes
1 answer
83 views

embed roots of unity from valued field into its residue field

Let $K_v$ non-archim valued complete local field with finite residue field $\kappa= \mathcal{O}_K / \mathfrak{m}_v$ of characteristic $p$. Assume $K_v$ contains $d$th roots of unity $U_d$. Is there ...
JackYo's user avatar
  • 179
1 vote
1 answer
61 views

Consistent Choice of Root of Unity in Characteristic $p > 0$

Let $\zeta_n$ be a primitive root of $x^n - 1$ in $\bar{\mathbb{F}}_p$ where $p \nmid n$. I'm assuming there's no canonical choice for $\zeta_n$ like in characteristic $0$, where we can just take $e^{...
Ryan Shesler's user avatar
  • 1,498
1 vote
0 answers
41 views

Find multiplicative function $f:\Bbb R\to$ $p$th roots of unity

For prime number $p$, find all functions $f:\Bbb R\setminus\{0\}\to\left\{\mathrm e^{\frac{2k\pi\mathrm i}p}\mid k\in\Bbb Z_+\right\}$ such that $f(ab)=f(a)f(b)$ for all real numbers $a$, $b$. This ...
youthdoo's user avatar
  • 1,475
0 votes
0 answers
60 views

Roots of Unity for fractional exponents

I am looking for a general solution for the roots of unity for a fractional exponent. That is to say, complex values of $z$, such that $z^n=1$ for $n=\frac{p}{q},p,q\in \Bbb{Z}, p$ is coprime to $q$. ...
Alex TJ's user avatar
0 votes
0 answers
53 views

Maximum absolute value of sum of m n-th roots of unity

Let $\omega, \omega^2,\dots,\omega^{n-1},1$ be $n$-th roots of unity, and consider $$S_n(m):=\max_{1 \le i_1 < i_2 <\dots<i_m \le n}|\omega^{i_1}+\dots+\omega^{i_m}|$$ i.e. the maximal ...
Peng Hao's user avatar
  • 153
1 vote
3 answers
83 views

Proving the identity $ \prod_{k=1}^{n-1}(1 - e^{\frac{2\pi ik}{n}}) = n$

I'm trying to prove this identity involving roots of unity, and would like to know if the following is a valid chain of reasoning. Looking at the complex polynomial solved by the n-th roots of unity ...
giorgio's user avatar
  • 583
0 votes
1 answer
100 views

Cyclotomic polynomial as minimal polynomial

I'm in the process of learning Galois theory and got stuck on Wikipedia's alternative definition of the $n$th cyclotomic polynomial as the "minimal polynomial over the field of the rational ...
kalanchloe's user avatar
0 votes
1 answer
46 views

How could I formalize this field extensions problem?

Let $\zeta_n$ be a primitive $n$th root of unity. I want to show that $F_n := \mathbb{Q}\left(\zeta_n\right)\cap\mathbb{R} = \mathbb{Q}\left(\zeta_n + \zeta_n^{-1}\right) = \mathbb{Q}\left(\cos\left(\...
user avatar
0 votes
1 answer
77 views

Root of Unity of any complex power

The problem is finding a complex number z, such that $$z^{a+bi} = z$$ I know for integer powers of z the Roots of Unity are $$z^{n} = e^{\frac{2πik}{n-1}}$$ I basically want to solve, given any ...
Dylan Woodbrey's user avatar
0 votes
0 answers
44 views

Ramanujan's sum & Roots of Unity

Ramanujan's sum is defined as: $$c_q(n)=\sum_{\substack{1\leq a\leq q \\\ (a,q)=1}}\exp\left(\frac{2 \pi ian}{q}\right)$$ Let $\eta_q(n)$ denote the sum of the nth powers of the qth roots of unity: $$\...
John Smith's user avatar
0 votes
1 answer
108 views

Permutating the coeffecients of equations with roots of unity.

I"ve noticed the roots of $$(x^k-1 =0) = (1-x^k=0)$$ This is simple enough since changing +/- signs gives you the additive inverse which is equal at zero. However this property can be also be ...
Older Amateur's user avatar
2 votes
0 answers
59 views

Calculating Coefficents of a single variable polynomial [duplicate]

Given: $$ (1+x+x^2+x^3+\cdots+x^k)^n $$ Is there a formula to calculate the coefficient of $x^a$ (where $a$ can be any integer value less than $k^n$) that's more efficient than grinding through ...
Older Amateur's user avatar
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0 answers
73 views

Algebraic forms for nth Roots of unity

Looking for algebraic forms for primitive nth roots of unity when n is composite. When n is prime it is simply all the possible roots of $ \sum_{m=0}^{\phi(n)} X^M =0 $ It's the composite values ...
Older Amateur's user avatar
-1 votes
1 answer
86 views

A Dilogarithmic Sum

I have been playing around with dilogarithm and considered the following sum. $$\sum_{n=1}^{k} {\text{Li}_{2} {(\omega^n)}}$$ Where $\text{Li}_2{(x)}$ is the Dilogarithm and $\omega$ is the k-th root ...
Alejandro Jimenez Tellado's user avatar
1 vote
0 answers
58 views

Root of unity and zero divisor

In a Wikipedia article on roots of unity modulo n, in the section on roots of unity, it is stated: If $x$ is a $k$-th root of unity and $x-1$ is not a zero divisor, then $\sum_{j=0}^{k-1} x^j \equiv ...
TreeBook1's user avatar
1 vote
0 answers
60 views

What is the cardinality of the set of $x$-th roots of unity?

Consider the set of $x$-th roots of unity. What is its cardinality? Formally, consider $x \in \mathbb{C}$, and let $S_x = \{y \in \mathbb{C} \; : \;$ there exist $k, m \in \mathbb{Z}$ such that $ x(\...
whoisit's user avatar
  • 3,227
0 votes
0 answers
71 views

$2$ as order $n$ unitroot in $\mathrm{GF}(2^n \pm 1)$: what should I know?

I'm currently trying to implement the DFT in Finite Fields in Schönhage Strassen matter as theoretical refreshment for the NTT (FFT variant of the NTT). My problem is that if I have an array of lets ...
rnnUSer11's user avatar
2 votes
1 answer
66 views

Factorizing Odd Prime Case of FLT using Roots of Unity [closed]

I am studying a fake proof of FLT which assumes $\mathbb Q\left[\zeta_p \right]$ is a UFD (that is, that the rationals with the pth root of unity is a UFD). We have covered the proofs for n=2, 3, and ...
dreadlearner's user avatar

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