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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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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Rational Trig Solutions for $n\ge 3$

Are there solutions to $$\sin(x+y)\sin(x-y)=n\ \sin(x)\sin(y)$$ for $n\ge 3$ where $x$ and $y$ are rational multiples of $\pi$? (excluding the trivial solutions when both sides are $0$). Known ...
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Sum involving roots of unity: $\sum \frac{1}{1-\omega^k}$

Let $\omega_n^k=\exp(2\pi ik/n)$ be the $n$-th roots of unity. I've come across the following sum a couple of times now (for example in a problem on hydrodynamics): $$ \sum_{k=1}^{n-1}\frac{1}{1-\...
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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 ...
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More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=6m+1\tag1$$ and the cubic, $$x^3+x^2-2mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
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Geometrical construction of 7th roots of unity given parabola x^2

Given parabola $x^2$ on plane, how can I construct 7th roots of unity? I was straying with my only idea, that sum of squares of real and imaginary part of roots of 1 equals 1 and belongs to $\mathbb{...
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When is a product of sums of roots of unity an integer?

A recent question asked for the residue modulo $101$ of $$A = \prod_{a=0}^9 \prod_{b=0}^{100} \prod_{c=0}^{100} (w^a + z^b + z^c)$$ where $w = e^{2\pi i/10}$ and $z = e^{2\pi i/101}$. I gave an answer ...
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Find $\sum\limits_{k=0}^{n-1}\,\omega_n^{k^2\ell}$, where $\omega_n:=\exp\left(\frac{2\pi\text{i}}{n}\right)$.

Let $n$ be a positive integer. Define the Gauss sum $$g(\ell,n):=\sum_{k=0}^{n-1}\,\omega_n^{k^2\ell}=\sum_{k=0}^{n-1}\,\exp\left(\frac{2\pi\text{i}k^2\ell}{n}\right)\,,$$ for every integer $\ell$...
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440 views

Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
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135 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
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Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
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How many tubes can you balance in a centrifuge?

I recently learned that if you have a centrifuge whose number of holes $n$ is divisible by $6$, then you can balance any number of tubes except for $1$ and $n-1$. If $k$, the number of tubes you want ...
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On the trigonometric roots of some quintics

I. Cubics The cubic $$x^3+x^2-2x-1=0$$ and its solution in terms of $\cos(n\pi/7)$ is well-known. A general formula for other primes $\color{blue}{p=6m+1}$ is discussed in this post. II. ...
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Automorphism(Galois groups) and galois theory

I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is: Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of $\mathbb{...
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284 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
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158 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
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For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed as ...
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root of a unit in a real biquadratic field

Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in $\mathbb{Q}(\...
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Let $w \in G_{24}$ be a primitive root. Find every $n \in \mathbb{N}$ such that $w^{15n} + \sum^{11}_{i=1} \overline{w}^{2i+6}$ is purely imaginary.

We know that $\overline{w} = w^{23}$. Given that $(24:23)=1$, $w^{23}$ is also primitive. But, for any such root $u$, $u^2 \in G_{12}$ is another primitive root. Hence, $$\sum^{11}_{i=1} \overline{w}^...
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Prove that $z_1,z_2,z_3$ with equal, non-zero modulus, are vertices of an equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $

Prove that $z_1,z_2,z_3 \in \Bbb C$ , distinct, with equal, non-zero modulus, are vertices of equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $ I tried dividing by $z_3|z_3|\...
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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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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 ...
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72 views

maximum distance between n points on unit circle

Let's n points $M_k$ on the unit circle. Find the position of these points such that $$P_n = \prod_{1 \le k \le l \le n}M_kM_l$$ (product of all the distances between the points) is maximum. With the ...
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76 views

Sum involving powers of roots of unity

For $a\in\mathbb{N}^*$ we define: $$ S(a)=\sum_{0\leq k \leq a-1} e^{i\pi \frac{k^2}{a}} $$ or equivalently, if $\omega=e^{\frac{i\pi}{a}}$ is the primitive $2a$-th roots of units, we could write: $$...
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86 views

Polygon discriminant sequence

The sequence A193681: Discriminant of minimal polynomial of $2*cos(\pi/n)$ has a lot of powers and almost-powers of $n$. Here's a picture. A135517 x n^(EulerPhi[2 n]/2 - 1) / A193681 gives 1, 1,...
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How to check whether $(x^n - 1)$ is an nth power?

I am looking for a way to check whether $(x^n - 1)$ is an $nth$ power when $x$ is rational and $n\in \Bbb N$. So far I found some solutions such as $(1.25^2 - 1)$ is $0.75^2$. I also found I can ...
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Linear Combination of Roots of Unity Equal to Zero?

Let $p$ be an odd prime number, and let $h\in\{1,\ldots,p-1\}$ be a primitive element of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$, i.e., $\{h^{k}\,(\mathrm{mod}\,p):1\leq k\leq p-1\...
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Evaluating a series of rational functions by hand or using generating functions

I'll jump right into the question: Let $\omega$ be a primitive $n$th root of unity. Show that \begin{align*} \frac{1}{n} \sum_{i=0}^{n-1} \frac{1}{(1-\omega^i t)(1 - \omega^{-i}t)} = \frac{1-t^{...
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Roots of Unity form a Real Algebraic Group

Does the group $\mu_n$ of $n^{\mathrm{th}}$-roots of unity form an algebraic group over $\mathbb{R}$ for every $n$? If so, why? I understand that $\mu_n$ forms an algebraic group over $\mathbb{C}$, ...
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(More) Explicit computation of roots of unity filter?

I am aware of the technique dubbed the "roots of unity filter" which calculates an expression of the form $\sum_{k|i}\dbinom{n}{i}$ in terms of the average of $(1+\omega^j)^n$ as $i$ and $j$ ...
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Direct algebraic manipulation on the roots of a polynomial?

I have a lot setup here but I want to provide some of the motivation for why I even have this question before I jump into the actual question. Motivation I have been working on trying to prove ...
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102 views

lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set $S = \{ |s(\vec x)| : \vec x \...
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479 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
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Textbook says roots of unity is equal to 1

The elements of the set $U_n = \{z \in \mathbb{C} : Z^n =1 \}$ are called the $n^{\text{th}}$ roots of unity. Using the technique of Examples 1.6 and 1.7, we see that the elements of this set are the ...
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What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
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upper bound on, or fast algorithm to find, an order $2^n$ element in the multiplicative group modulo prime $q(2^n)+1$

I have a program which (in its current implementation) requires, for a given $N=2^n$, some $\omega$ in some field such that $\omega^N=1$ and $\omega^i\ne1$ for each $0<i<N$. Complex roots of ...
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About sums similar to gauss sums

It is well known the case for sums like: $$ \sum_{i=0}^{p^n -1}\zeta^{-ai}, $$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$ \sum_{i=0}^{p^n -1}i^...
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Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
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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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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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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 ...
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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 ...
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Question about proof regarding cyclotomic polynomials from Dummit & Foote

I'm currently reading through Dummit & Foote's Abstract Algebra textbook. I'm in section 13.6, Cyclotomic Polynomials and Extensions, which is focused around proving that $[\mathbb{Q}(\zeta_n):\...
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71 views

Property of roots of unity

Let $R$ be a commutative ring and $\zeta$ be a root of unity in $R$ such that $\mathrm{ord}(\zeta) = n$. If $n = p_1^{e_1} \dots p_r^{e_r}$ is the decomposition in prime factors of $n$, is it true ...
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33 views

Closed-Form Expression for a Certain Sum Over Roots of Unity

Let $p$ be an odd integer $\geq3$, let $n$ be an arbitrary positive integer, let $c$ be a non-zero complex number, and let $\omega$ be a complex variable. Does anyone know of a closed-form expression ...
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384 views

Roots of unity and vandermonde matrix

It says we define $w={\normalsize e}^{\Large{\frac{2\pi i}{N}}}$, the $N^{th}$ root of unity (that is $w^N=1$ ) also, $\bar{w}$ is the complex conjugate of $w$ that is since $w={\normalsize e}^{\...
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50 views

About $n$'th rooth of $-1$ in a finite field

This question is based on this other question from @Malkoun, and more specifically the comments. Context. Let's recall the following proof which shows that $-1$ is a square if $\mathbb F_p$ for $p\...
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Is there a term for a set of eigenvalues not containing any roots of unity except 1?

Is there a term describing a matrix $A \in \mathbb C^{n\times n}$ whose eigenvalues are all either $1$ or not a root of unity? Is there a term for a holomorphic selfmap $F$ of $\mathbb C^n$ fixing $0 ...
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118 views

Distinct elements based off roots of unity

I'm working on a problem, but I haven't been able to get much out of it... Given that $\zeta$ is a primitive complex n-th root of unity, show that the complex numbers $1,\zeta,\zeta^2,...,\zeta^{n-1}$...