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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(\...
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What will be the number of distinct element in the set of the roots of unity?
Let $\omega$ denote a cube root of unity which is not equal to $1$. Then what is the number of distinct elements in the set
{${(1+\omega+\omega^{2}+\cdots+\omega^{n})^{m}:m,n=1,2,3\cdots}$}
My answer ...
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$2$ 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 ...
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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 ...
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Sum of a pth root of unity raised to square powers
If $p$ is prime and $S = \sum_{j=0}^{p-1} {\varepsilon}^{j^2}$, where $\varepsilon$ is a primitive $p$th root of unity, what is $S^2$?
What I did so far is write $S^2 = \sum_{0 \le a,b \le p-1} {\...
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Bounding a Real Function in Absolute Value
Show that if $\omega$ is a primitive $2m^\text{th}$ root of unity, where $m\geq 1$ is an integer, then for all $i\geq 0$ and $x\in [0,1]$, we have the inequality $$\prod_{j=0}^{2m-1}|1-x^{(2i+1)m+j}\...
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Do all of these partial sums of roots of unity have real part = $\frac{1}{2}$ except at powers of $2$?
Let the Dirichlet inverse of the Euler totient function be:
$$\vartheta(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
and compute the sum:
$$q(x,n)=\sum _{h=0}^{\infty } \left(\sum _{k=1}^n x^{h n+k}...
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Two questions about Dirichlet's characters and roots of unity.
Let $G$ be the units of $\mathbb Z/ q \mathbb Z.$ Let $\chi$ denote Dirichlet character on $G.$ I have two questions.
Let $a \in G$ with order $k, $Then it is easy to see that $\chi(a)$ is a $k$ th ...
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Trying to understand this skipping sum
$$ f(x) = \begin{cases} 0 & x \not\equiv 4 \pmod 5 \\ 1 & x \equiv 4 \pmod 5 \end{cases} $$
and $$ f(x) = \frac{1}{5} \sum_{k=0}^4 \cos\left(\frac{2 \pi}{5} k (x-4) \right) $$
Can someone help ...
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Finding the number of distinct common roots of unity
Consider equations:
$${x}^{p} = 1\\
{x}^{q} = 1$$
Then the number of common roots is equal to the $\gcd(p,q).$
But, how can I prove this statement?
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$(\zeta_n + 1 + \zeta_n^{-1})(\zeta_n^a + 1 + \zeta_n^{-a})=1$.. Possible solutions when $(a,n)=1$.
$(\zeta_n + 1 + \zeta_n^{-1})(\zeta_n^a + 1 + \zeta_n^{-a})=1$ where both $\zeta_n$ and $\zeta_n^a$ are different primitive $n$-th roots of unity such that the terms are distinct (i.e. $a\ne\pm1$). ...
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Coefficients of a symmetric product of polynomials with root of unity
For number $n\ge2$, let $\xi$ be a primitive $n$-th root of unity.
The determinant of circulant matrix is a symmetric polynomial in $c_0,\dots,c_{n-1}$
$$f_n=\prod_{j=0}^{n-1}\sum_{i=0}^{n-1}ξ^{ij}c_i$...
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Lattice Geometry of $\mathbb{Z}[\zeta_5]$
I was trying to plot all the points of $\mathbb{Z}[\zeta_5]$ and see if there is a nice lattice structure.
It is easy for the Gaussian Integers: $\mathbb{Z}[\zeta_4] = \mathbb{Z}[i]$, which is a ...
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What sums of roots of unity can be expressed "simply"?
Given $\zeta_n = e^{\frac{2i\pi}{n}}$ for natural $n$,
When is $S = \zeta_{n_0}^{m_0} + \zeta_{n_1}^{m_1} + \dots = M\sqrt[r]{R}^k\zeta_{n}^m$, for rational $M, R, m$ and natural $r, k$?
So far I ...
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Intermediate Extensions of $\Bbb Q[\zeta_{63}]/\Bbb Q$.
Let $\zeta$ be the $63^{th}$ root of unity. The reason why I choose the number $63$ is because $$\operatorname{Gal}(\Bbb Q[\zeta]/\Bbb Q)\simeq (\Bbb Z/63\Bbb Z)^\times\simeq(\Bbb Z/2\Bbb Z)\oplus(\...
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Given $\theta$, how to check if $e^{i\theta}$ a root of unity?
I have edited my question since I clearly wasn't communicating well what I was looking for.
I am given $\theta$ such that $e^{i\theta}$ is a root of unity. We require
$$\theta=q\pi,\quad q\in\mathbb{Q}...
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Prove $\prod_{k = 1}^{n - 1}(x - e^{\frac{2\pi ik}{n}}) = \frac{x^n - 1}{x - 1}$
I'm reading the solution to a math question, and part of the solution states this:
$\prod_{k = 1}^{n - 1}(x - e^{\frac{2\pi ik}{n}})$ is equal to
$\frac{x^n - 1}{x - 1}$ by the roots of unity.
...
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IMO proposal question. prove $\sum_{k=1}^{n} k \cos(\frac{2 \pi a_k} n) = 0$
I was looking into the problems from the art of problem solving by paul zeitz. I was stuck with the following question.
Let $n$ be a positive integer having at least two distinct prime factors. Show ...
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Number of Distinct Roots of Unity
The roots of unity are the solutions to:
$$z^n=1,z\in \mathbb{C},n\in \mathbb{N}\implies(rcis\theta)^n=1\implies r=1,\theta=\frac{2\pi k}{n},k\in\mathbb{Z}$$
A primitive root of unity for $n$ is a ...
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$P(z)=4z^4+az^3+bz^2+cz+d$ where $0\le d\le c\le b\le a\le 4$ has a zero $z_0$ of magnitude $1$. Find the sum of $P(1)$s of all such polynomials $P$.
Here's the problem statement:
Consider all polynomials of a complex variable,
$P(z)=4z^4+az^3+bz^2+cz+d$, where $a,b,c,$ and $d$ are integers, $0\le d\le c\le b\le a\le 4$, and the polynomial has a ...
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Roots of Unity Sum Proof
If $1,\omega,\omega^2,...\omega^{n-1}$ are complex numbers such that $\omega^n-1=0$, then
they are the $n^{th}$ roots of unity
they lie on the vertices of a regular polygon on the unit circle with ...
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Doubt in Cyclotomic Fields and polynomials
I am reading A Classical introduction to Number Theory by Ireland Rosen. There, they have the following statement and proof:
Let $K/\mathbb{Q}$ be an algebraic number field and let $\sigma_1, \...
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length of side of a regular $n$-gon is less than length of any diagonal
In any regular polygon with $n\ge 4$ sides, why is any side length strictly length to any diagonal length? (A diagonal is defined as the line segment joining non-adjacent vertices)
This is intutitvely ...
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Zeros of a complex function on the unit circle
I need a closed form for the zeros of $$f(z):=\frac{z^{2(1+\omega)}-z^{2\omega}}{i}-e^{-\frac{\pi}{2\sqrt{3}}} $$ where $z$ lies on the unit circle $|z|=1$ and $\omega$ is a cube root of unity.
We ...
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Is there an element of order $n$ in all roots of $x^n - 1$ modulo $q$ where $q$ is a prime number and $n \mid q-1$
In the field of $\mathbb{Z}_q$ where $q$ is a prime, if $n \mid q-1$, then
$$
x^n - 1 = (x - \omega_1) (x - \omega_2) \cdots (x - \omega_n)
$$
where $\omega_i$ is a root of $x^n-1$ for all $1 \leq i \...
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Computing a sum involving the roots of a tenth degree polynomial
Let $$p(x)=\sum_{k=0}^{10}x^k=1+x+\ldots+x^{10}.$$ Let the roots of $p$ be $\alpha_i,i\in\{1,2,\ldots,10\}.$ Compute $$\sum_{i=1}^{10}\frac{1}{1-\alpha_i}.$$
My Attempt: Using the GP formula, we get ...
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If $f(x)$ is a monic polynomial in ${\mathbb Z}[x]$ and all roots have absolute value 1 , then all roots are roots of unity.
I am trying to understand the statement that was mentioned here.
If $f(x)$ is a monic polynomial in ${\mathbb Z}[x]$ and all roots have absolute value 1, then all roots are roots of unity.
I was ...
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If $\sin(\pi/13)\sin(3\pi/13)\sin(4\pi/13)={a\over b}\sqrt{\frac{k-3\sqrt k}{2}}$, find $\frac{5a+b}{k}$.
If $\sin(\pi/13)\sin(3\pi/13)\sin(4\pi/13)={a\over b}\sqrt{\frac{k-3\sqrt k}{2}}$, find $\frac{5a+b}{k}$.
Let $s=\sin(\pi/13)\sin(3\pi/13)\sin(4\pi/13)$ and $x=\sqrt k$. Then the above equation ...
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Proving the locus of $P_k$ is an ellipse.
I was recently working on the following problem functioning within an overlap of complex numbers and coordinate geometry:
Let $z$ be a complex number $a + ib$ (where $a > b > 0$), and $α_k$ ($0 &...
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Schonhage-Strassen algorithm for multiplication of polynomials over a finite field (additive vs multiplicative complexity)
Trying to understand the Schonhage-Strassen algorithm for multiplying two polynomials $f(X)$, $g(X)$ of degree $n$ over a finite field $\mathbb{F}_q[X]$ with $q$ a prime such that $q-1$ does not have ...
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Any unit in $\Bbb Z[\zeta_p]$ can be decomposed into a power of $\zeta_p$ and a real unit in $\Bbb Z[\zeta_p]$
I am trying to prove the following result, which states that any unit in $\Bbb Z[\zeta_p]$ can be (multiplicatively) decomposed into a power of $\zeta_p$ and a real unit in $\Bbb Z[\zeta_p]$.
Let $K =...
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Double factorial & roots of unity filter
Original problem statement:
For any positive integer $n$, let $(2n)!!$ be the product of all positive even integers less than or equal to $2n$, By convention, $0!!=1$.
For example, $6!!=6\cdot4\cdot2=...
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Significance of Roots of Unity
A strong precalculus course (think AMC, JEE) will teach complex numbers, specifically $n^{\text{th}}$ roots of unity with their properties:
They lie equally spaced on the unit circle
A power of an $...
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Prove all Lie group homomorphisms of the circle have certain form
Prove all lie group homomorphisms $\phi:\mathbb{S}^1\to\mathbb{S}^1$ has the form $z\to z^n$ for some $n\in\mathbb{Z}$.
My idea - first of all, I cannot use the Lie algebra-lie group correspondence ...
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What is known about the minimal absolute value in $\mathbb{Z}[\zeta_n]\setminus\{0\}$?
Here $\mathbb{Z}[\zeta_n]$ is the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the $n$th root of unity. In my mind $\mathbb{Z}[\zeta_n]$ looks like a grid in the ...
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Why does there seem to be a relation between the fifth roots of $32$ and $\varphi$ (the golden ratio)?
I have the following problem on one of my assignments:
find all five fifth roots of 32.
What I find interesting is how $\varphi$ is related to the real component of most of the solutions. My ...
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Why is my equation not true on all elements?
Consider a finite field $F_{17}$. In this, $\omega = 4$ is the primitive $4$th root of unity.
So there is a subgroup $\Omega = \{1, \omega, \omega^2, \omega^3\}$
Consider a polynomial $f \in F_{17}[x]$...
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Prove that $yx = qxy$ implies $(x + y)^d = x^d + y^d.$
Here is the question I am trying to prove:
Let $q\ne1$ be a root of unity of order $d > 1.$ Prove that $yx = qxy$
in a noncommutative algebra implies $$(x + y)^d = x^d + y^d.$$
I do know how to ...
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Product of differences of roots of unity
$\DeclareMathOperator{\lcm}{lcm}$
Denote by $\mu_n$ a primitive n-th root of unity. Examples suggest that
$$\prod_{0\le k\lt a \\ 0 \le l \lt b \\ k/a \neq l/b} (\mu_a^k - \mu_b^l)=\pm \lcm(a,b)^{\gcd(...
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Is every complex root of an integer polynomial a product of an algebraic real number and a root of unity?
If $f \in\mathbb{Z}[x]$ and $u \in \mathbb{C}$ is a root of $f$, then do we always have that $u = a\xi$ where $a$ is some real algebraic number, and $\xi$ is some root of unity?
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Norm of a root of unity $\zeta$ is $1$ only if $\zeta=1$?
Let $p$ be an odd prime number. Let $K/\mathbb Q_p$ be a finite extension, with $K$ having residue field $\mathbb F_q$ of order $q$ some power of $p$. Let $\zeta\in\mu_{q-1}\subset K$ be a $(q-1)$st ...
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Integer polynomials such that $f(z) = 0, |z| = 1$
Let $f(z)$ be a polynomial with integer coefficients of degree $n$.
Also $f(z)$ is irreducible over the integers, and it is not a cyclotomic polynomial.
Let $v$ be a given nonreal complex number.
...
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Show that $\sum_{j=0}^{n-1} {|{z_1} + {\omega}^j{z_2}|}^2 = n(|{z_1}|^2 + |{z_2}|^2)$
If $\omega^j, j=0,1,2,...,n-1 $ are the $n^{th}$ root of unity, show that $$\sum_{j=0}^{n-1} {|{z_1} + {\omega}^j{z_2}|}^2 = n(|{z_1}|^2 + |{z_2}|^2)$$ for any two complex numbers ${z_1}$ and ${z_2}$.
...
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How do unity roots relate to the summation?
let $w_0,...,w_{2n}$ be the $(2n+1)$-th root of unity. Compute $$S=\sum\limits_{k=0}^{2n}\frac{1}{1+w_k}$$
I tried this many times but I just can't see how it's done.
My idea is simplify the function ...
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Multiplicative complex function has mean value
I came up with the following question, and I don't have proof of it.
Let $m>1$ be a positive integer. Let $f:\mathbb{N}\to \mathbb{C}$ a multiplicative function, whose image is a subset of the $m$-...
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