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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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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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2answers
55 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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25 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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40 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
27 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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37 views

$a_i$ 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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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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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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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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46 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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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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1answer
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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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
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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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83 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
43 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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1answer
92 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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1answer
43 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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$ \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
46 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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1answer
293 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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113 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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2answers
105 views

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
96 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
46 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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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 ...
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1answer
59 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
57 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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331 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 ...
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1answer
76 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
69 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
109 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
25 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 ...
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1answer
44 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
36 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
234 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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1answer
43 views

Find a polynomial in $\mathbb{Z}_{41}$

Find a $7^\text{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}[x]$, so that $$ p(14^i) = i\pmod{41}\ \forall i = 0,1,\ldots,7. $$ Hint: $3$ is the $8^\text{th}$ primitive root of unity and $3 \...
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3answers
46 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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1answer
52 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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1answer
66 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
77 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
17 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 ...