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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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In general, do primes of the form $a×2^{n}+1$always have primitive $2^n $th roots of unity (modulo that prime)?

EDIT: Title had an extra +1 in the 2's exponent For context, in competitive programming, problems which require a number theoretic transform usually ask for the answer modulo $998244353=7\times17\...
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1answer
39 views

Sub-Sum of Roots of Unity

Let $\alpha$ be an algebraic integer of a cyclotomic field, and let $\theta_1, \theta_2, ..., \theta_n$ be roots of unity such that $$\sum_{i=1}^n \theta_i = 2\alpha.$$ Does there necessarily exists a ...
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22 views

Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
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1answer
37 views

What online graphing tools handle complex numbers well?

What online graphing tools handle complex numbers well? Desmos is generally excellent by breaking functions down into their real and imaginary parts and plotting on the Euclidean plane. For example ...
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1answer
64 views

Question about roots of unity in the Fast Fourier Transform

I am learning about the Fast Fourier Transform, which converts a polynomial from its coefficient representation into its point-wise form using divide-and-conquer. The Fast Fourier Transform evaluates ...
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1answer
63 views

Finding minimal polynomial of generator of quadratic subfield of cyclotomic extension $\mathbb{Q}(\xi_{17})$

I will prove that $K_2=\mathbb{Q}(\sqrt{17})$ skipping a few 'trivial' steps to keep the post short. There is a theorem that says that Gal$(\mathbb{Q}(\xi_{17}))\cong\left(\mathbb{Z}_{17}\right)^{\...
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47 views

If $A^{2016} = I_n$, show that $A^{576} - A^{288} + I_n$ is invertible, and calculate it's inverse in terms of $A$.

Let $A$ be a real valued $n \times n$ matrix,where $n \geq 2$, such that $A^{2016} =I_n.$ Show that the matrix $B = A^{576} - A^{288} + I_n$ is invertible, and calculate it's inverse in terms of $A$. ...
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45 views

Cyclotomic polynomials of distinct index are distinct.

Does there exist two cyclotomic polynomial $\Phi_n$ and $\Phi_m$ which are equal but $n\neq m$? The cyclotomic polynomial is defined as $\Phi_n(x)=\prod_{\substack{1\le j\le n \\ \gcd(j,n)=1}}(x-u_{(...
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28 views

Roots of unity divisibility.

Suppose $r | n$. Then $R:= e^{2i \pi k/r}$ is an $n$-th root of unity. Thus, there exists a unique $l \in \{0, \dots, n-1\}$ such that $R = e^{2\pi i l/n}$. Does it hold that $l |n$? I tried to ...
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If $a \in \mathbb{C}$ and $\exists n \in \mathbb{N}$ s.t. $\{ a^n, a^{n+1} \} \in \mathbb{N}$, prove $a \in \mathbb{N}$ [closed]

Let $a$ be a complex number. If it exists a natural number $n$ (different of $0$), such that $a^n$ and $a^{n+1}$ are integers, prove that $a$ is an integer.
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46 views

Do there exist non-trivial integer coefficients that break linear independence of the roots of unity?

Let $n$ be a positive integer, $k=0,\cdots,n-1$, $\omega_k=e^{\tfrac{2\pi i}{n}k}$ be the roots of unity, $c_k \in \mathcal{Z}$ be integer coefficients, trivial and non-trivial be two subcategories ...
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60 views

To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$

Sum the series: $$ \frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!}, $$ $n$ being a multiple of $3$.(Math. Tripos, 1899) My attempt We may ...
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26 views

If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

A lemma in my lecture notes states that if $K$ is a subfield of $\mathbb{C}$ and $\zeta=\exp(2\pi i/p)$ then $K(\zeta)/K$ is Galois. They proved it by arguing that the minimal polynomial of $\zeta$ ...
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If $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$.

How do I prove that for all natural numbers $n$ and complex numbers $a, b, c, z, w$ if $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$. ...
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degree of extension re: roots of unity to the $2^n$ power and their conjugates

Let $\zeta_{2^{n+2}}$ be a $2^{n+2}$th root of unity, and let $\overline\zeta_{2^{n+2}}$ be its complex conjugate. I am looking for help in showing that $[\mathbb{Q}(\zeta_{2^{n+2}}): \mathbb{Q}(\...
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1answer
45 views

Another Roots of Unity Sum

I almost see a brute-force attack on this problem, but before messing with the details I wonder there is some theory here, or at least a nice way to group the terms so I can see the cancellation. Let ...
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1answer
28 views

Direct product decomposition of the group of complex roots of unity

I'm studying $p$-adic numbers (Robert's "A course in $p$-adic analysis) and, at page 41, the author states that, for every prime $p$, the group $\mu$ of all complex roots of unity has a direct product ...
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$a_i$ are the n-th roots of $1\in\mathbb{C}$, why does $(1-a_2)\cdot…\cdot(1-a_n)=n$?

For $1<i\leq n$, let $a_i$ be the n-th roots of $1\in\mathbb{C}$, why does $(1-a_2)\cdot...\cdot(1-a_n)=n$?
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Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}$

Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}.$ I'm trying to use Dirichlet's Unit Theorem to solve this problem. It states that if $K$ is a number ...
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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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Computing the sum of inverses of some roots of 1 in a field, given their sum

Fix an algebraically closed field $F$. Let $\alpha_1,\dotsc,\alpha_n\in F$ be roots of $1$. Let $x=\alpha_1+\dotsc+\alpha_n$ and $y=\alpha_1^{-1}+\dotsc+\alpha_n^{-1}$. I was thinking: Given $n$ and ...
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24 views

Sign of a linear combination of roots of unity

Is there a way to access the sign of an integer, self conjugate, linear combination of roots of 1? More precisely, is there an algorithm (fast is preferred :-) that, given rationals $q_1,q_2,\ldots,...
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1answer
50 views

Degree of splitting field of $X^n-1$ over some finite field

Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the ...
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206 views

If $ \alpha_i, i=0,1,2…n-1 $ be the nth roots of unity, the $\sum_{i=0}^{n-1} \frac{\alpha_i}{3- \alpha_i}$ is equal to?

If $ \alpha_i, i=0,1,2...n-1 $ be the nth roots of unity, the $\sum_{i=0}^{n-1} \frac{\alpha_i}{3- \alpha_i}$ is equal to? A) $ \frac{n}{3^n-1} $ B) $ \frac{n-1}{3^n-1} $ C) $ \frac{n+1}{3^n-1} $ D) ...
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82 views

Series involving complex roots: $\frac{1}{2-a_1} + \frac{1}{2-a_2} + \dots + \frac{1}{2-a_{n-1}} = \frac{(n-2)2^{n-1}+1}{2^n - 1}$

$$ \frac{1}{2-a_1} + \frac{1}{2-a_2} + \dots + \frac{1}{2-a_{n-1}} = \frac{(n-2)2^{n-1}+1}{2^n - 1} $$ Here $1,a_1,a_2,\dots,a_{n-1}$ are $n$-th roots of unity I know the sum of roots is 0. I think ...
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114 views

Series Summation: $\sum\limits_{k=1}^{N-1}\frac{1}{z-w_k}$ where $w_k=e^{\frac{2\pi i k}{N}}$

I have the series $$\sum_{k=1}^{N-1}\frac{1}{1-w_k} $$ where $w_k=e^{\frac{2\pi i k}{N}}$, how can I find the summation of this , another question related to this sum $$\sum_{k=1}^{N-1}\frac{1}{z-w_k} ...
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Find $\sum_{i=1}^{n-1}\frac{1}{2-a^i}$ if $1,a,a^2,…,a^{n-1}$ are the n$^\text{th}$ roots of unity [duplicate]

If $1,a,a^2,...,a^{n-1}$ are the n$^\text{th}$ roots of unity, then prove that$$\sum_{i=1}^{n-1}\frac{1}{2-a^i}=\frac{(n-2)2^{n-1}+1}{2^n-1}$$ $$ \alpha_r=e^{i\tfrac{2\pi r}{n}}=a^{r-1}\\ x^n=1\...
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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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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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1answer
127 views

Geometry using Complex Numbers/Roots of Unity [duplicate]

I have no idea how to solve this problem, but I'm pretty sure that it could be made easier by using Roots of Unity! Help would be appreciated! Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon ...
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1answer
85 views

Why are roots of unity evenly spaced?

Roots of unity are the solutions of the complex polynomial $t^{n}-1=0$ they have the following form $E_{n}=\{e^{\frac{2\pi ik}{n}}:k\in\mathbb{Z}\}=\{e^{\frac{2\pi ik}{n}}:k=1,...,n-1\}$. From the ...
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1answer
45 views

Kummer ring - special monic polynomial with zero at root of unity

Let $ \zeta_n = e^{2 \pi i / n} $ be the n-th root of unity. Let $$ P(z) = \sum_{k = 0}^{n-1} s_k z^k $$ be a monic polynomial over $ z \in \mathbb{C} $, specified by integer coefficients $ s_k \...
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121 views

(Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Here is the work I've done so far: $\sqrt[p]{2}$ is a real root of $f(x)$ Any $(\zeta \sqrt[p]{2})$ where $\zeta$ is a $p^{th}$ root of unity is also a root of $f(x)$ Since $p$ is prime, all its ...
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45 views

Why is $\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$ independent of $\alpha$?

By accident I found (numerically) that the expression $$\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$$ only depends on $\beta$. This looks like it shouldn't be ...
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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 ...
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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 ...
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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 ...
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3answers
118 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\} $
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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 ...
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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^\...
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1answer
50 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{...
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53 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
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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 $w_1$ ...
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3answers
65 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?
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1answer
357 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
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 ...
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1answer
72 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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30 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
111 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
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 ...