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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1answer
35 views

Product of Differences of nth Roots of Unity

I'm trying to show that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=n$$ but am finding it surprisingly difficult. I know by symmetry that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ...
2
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1answer
25 views

Splitting field of $f=X^p -a \in \mathbb{Q}[X]$.

Let $p$ be a prime number, and $a \in \mathbb{Q}$, a number such that there is no integer $k$ satisfying $p^k=a$. Write $f= X^p -a \in \mathbb{Q}[X]$. I have to prove the following statements: The ...
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0answers
25 views

Isomorphism betwixt a Galois Group and the set of $n$th roots of unity

Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...
6
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1answer
92 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 know ...
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0answers
19 views

About the rank of (sub) matrices whose entries are roots of unity

Let $\Omega$ be a matrix with entries $a_{jk}=\omega^{jk}$, where $0\leq j,k\leq n-1$, and $\omega=e^{-2\pi i/N}$, with $N\in \mathbb{N}$, so $\Omega$ looks like $$ \Omega=\begin{pmatrix} 1 & 1 ...
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0answers
20 views

Summing complex numbers of magnitude $1$

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 ...
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0answers
17 views

r a primitive root of unity show, (r-1)/((r^k) - 1) is an algebraic integer in Q(r)

I left out some hypotheses in the title to keep things short, so here is the full form: Let r be a primitive mth root of unity for m>1 and let k be a positive integer such that gcd(m,k)=1. Show (r-1) ...
1
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1answer
42 views

Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
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1answer
148 views

Proof with roots of unity

Let $m,n \in \mathbb N$ and $d=gcd(m,n).$ Prove that if w is both an m-th root of unity and an n-th root of unity, then w is a d-th root of unity. How would i begin about starting this type of proof? ...
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0answers
32 views

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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3answers
68 views

Showing that $\mathbb{Q}(\zeta_p, \sqrt[p]{\ell}) = \mathbb{Q}(\zeta_p + \sqrt[p]{\ell})$ for $p,\ell$ primes.

We consider the polynomial $x^p - \ell$, where $p,\ell$ are both prime numbers. Let $\zeta_p$ be a $p$-th root of unity. We wish to show that $L = \mathbb{Q}(\zeta_p, \sqrt[p]{\ell})$ is the same as ...
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1answer
32 views

Express $\cos(\pi/5)$ in terms of a sum of powers of the principal 100th root of unity.

Express $\cos(\pi/5)$ in terms of a sum of powers of the principal $100{th}$ root of unity. Using the formula, $w_n = \cos(2\pi/n) + i \sin(2\pi/n)$ I have calculated, $w_{100} = \cos(\pi/50) + i ...
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0answers
24 views

Complex Roots of Unity - In terms of $\sin$ and $\cos$ what is $w=w_{100}$, the principal $100^{th}$ root of $1$

I've been assigned this question on Complex Roots of Unity. In terms of $\sin$ and $\cos$ what is $w = w_{100}$ the principal $100^{th}$ root of 1? The $100^{th}$ root of unity is $w_n = \cos(2pi/n) ...
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4answers
38 views

Solving an equation for an unknown

How can I solve the following equation for $q$? I'm totally stuck. I have done everything up to this point though. $$\left(q + \sqrt{q^2-1}\right)^{2(N+1)} = 1,$$ where N is a natural number. ...
3
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3answers
70 views

Roots of unity modulo $p$

Let $f(X)$ be the minimal polynomial of something like $\zeta + \frac{1}{\zeta}$, where $\zeta$ is a primitive $p$-th root of unity for some prime $p > 2$. I'd like to show that $f(X) \equiv ...
3
votes
1answer
50 views

Cyclotomic Character

I have a couple of questions concerning the cyclotomic character. For the moment I know very little about the mod $\ell$ cyclotomic character, namely that ...
0
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1answer
58 views

Finding the qth root of unity mod p

I have $p$ and $q$ as $p = 4916335901, q = 88903$ and I have to find the $q^{th}$ root of unity $\pmod{p} $, so its $q{th}$ root of unity $\pmod{4916335901}$. What exactly is a $q^{th}$ root and what ...
0
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1answer
53 views

Roots of Unity of a specific argument

I am asked to find an unstable period 5 point for $f(z)=z^2$ with an argument which lies between -0.74 and -0.44. I can solve to get all the roots of unity, but how can I narrow it down the the one ...
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0answers
58 views

May someone tell me where I went wrong when finding a closed-form solution to a recursion?

Let $r(n)=\left\lceil\frac{n}{4}\right\rceil$, where $k \in \mathbb{N}$. Note that this can be rewritten as a recursion $R(n)=R(n-4)+1$, where, $R(0)=0, R(1)=1, R(2)=1, R(3)=1$. Since this recursion ...
1
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1answer
27 views

Problem with the proof that galois extension of $x^n-1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}_n)^\times$

I'm trying to understand a proof that galois extension of $x^n-1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}_n)^\times$. I can see why there is an injection of $\text{Gal}(x^n-1)$ in ...
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4answers
60 views

Proof roots of unity being in $\mathbb R$

Let $n \in \mathbb N$ even, and be $w,z \in \mathbb G_n$ primitives. Proof that $(w+z)^{n/2} \in \mathbb R$. Ok, as I didn't really know how to start, I tried several things, such using the Binomial ...
1
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1answer
42 views

Question on $2^N$th Roots of Unity within a function.

Prove that, if $w$ is a $(2^N)$th root of unity, where $N \in \mathbb N$, then: $$\lim_{r\to 1^-}|f'(rw)| = \infty$$ Where: $$f(z) = \sum\limits_{j = 1}^\infty 2^{-j}z^{2^j}$$ I haven't done left ...
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3answers
115 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
0
votes
1answer
42 views

How large is the largest prime required to satisfy these requirements?

I require a set of primes, all being equal to or greater than $2v+2$. The product of the primes should be at least $(2^v)+1$. I have one additional constraint. Each prime minus one must be ...
3
votes
1answer
88 views

Properties of a sum over the root-of-unity expression of polynomials over a finite field

Consider a bivariate polynomial over the finite field $\mathbb{Z}_n$ of the form: $$f(x,y) = c\cdot xy + g(y)$$ where $c$ is some non-zero constant and $g$ is some univariate polynomial. Let ...
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1answer
31 views

Roots of unity product

For each $n \in \mathbb N, n \geq 3$ calculate the product of all the n roots of unity. Or to say it in a more stric way: $$\prod_{w \in G_n^*}w$$ Being $G_n^*$ the primitive roots of the unity.
0
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1answer
35 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
0
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1answer
29 views

Exercice whith primitive roots of unity and divisibility

For $n \in \mathbb{N}$, we define $\Phi_n \in \mathbb C[x]$ as the monic polynomial that has as roots the $n$th primitive roots of the unity. For example $\Phi_2 =(x+1)$, $\Phi_4 = (x-i)(x+i) = ...
3
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1answer
39 views

$p$-th roots of unity adjoined to a $\mathfrak{p}$-adic field

I want to prove the following: Let $k$ be a number field and $S$ a set of primes of $k$ containing the primes $S_p$ that lie over the rational prime $p$. Then the extension of $k$ by the group of ...
1
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2answers
74 views

Roots of Unity - Complex Numbers

The sets $A = \{z : z^{18} = 1\} $and $ B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. ...
0
votes
1answer
116 views

Precalculus - Complex numbers/Roots of Unity/Exponential Form

Find the roots of $6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 = 0.$ Erm...it sorta looks like $(x+y)^6 = x^6 +6x^5y + 15x^4y^2 + 20x^3y^3 + 15y^4x^2 + 6y^5x + y^6$ Any idea how to proceed from here? ...
1
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3answers
98 views

Complex numbers and Roots of unity

I have no clue how to begin these problems. How do I start? I don't think I should pound em out...Thanks. Let P be the set of $42^{\text{nd}}$ roots of unity, and let Q be the set of $70^{\text{th}} ...
0
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0answers
18 views

Sum of roots of unity bounded away from 0

Let $n\in \mathbb N$ and $\zeta_N$ be a primitive $N$th root of unity. Let $a_k\in \mathbb Z,0\le k<N$. Assuming that the sum is nonzero, find a lower bound on the absolute value of $$ ...
3
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5answers
383 views

I'd like to get explain about complex roots

If $x^6+1=0$ so $x^6=-1$, then we have to find the roots at $\mathbb{C}$. I saw that the roots are $$\Large{e^{(\frac{\pi}{6}+\frac{2k\pi}{6})i}}\;\small{k=0,1,2,3,4,5}$$ this what I understand. ...
1
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1answer
69 views

Roots of unity and a system of equations by Ramanujan

Is it immediately apparent that the solution to the system of equations, $$\begin{aligned} x_1^2 &= x_2+2\\ x_2^2 &= x_3+2\\ x_3^2 &= x_4+2\\ &\vdots\\ x_n^2 &= x_1+2\\ ...
0
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2answers
128 views

Proving equations involving the powers of a complex cube root of unity ω

The question in this homework problem is to show $ω^4 + ω^5 = -ω^6$ given that $ω$ is a complex cube root of unity. I am also required to show that $(1 - ω)^2 = -3ω$, but if I am assisted with ...
1
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4answers
52 views

Proof that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}$

Being $\mathbb G_n$ the roots of unity for $n \in \mathbb N$, prove that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}.$
3
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2answers
129 views

How can I find the fifth root of unity?

I need to find fifth root of unity in the form $x+iy$. I'm new to this topic and would appreciate a detailed "dummies guide to..." explanation! I understand the formula, whereby for this question I ...
1
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1answer
24 views

Prove Bijection in roots of unity function

Given $k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} $. Probe that if $n$ and $m$ are coprime, the function $f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta$ is bijective. ...
0
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0answers
49 views

Conjecture about some rings and roots of unity. [duplicate]

Let $\Bbb R_{\geqslant 0}[X_n]$ be a polynomial semiring. More precisely $\Bbb R_{\geqslant 0}[X_n]$ are the polynomials of $X_n$ with positive real coefficients with $(X_n)^n = 1$. Let $F(n)$ be ...
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0answers
42 views

Roots of unity in CM-field

Let $K$ be a CM-field, ie. a totally imaginary quadratic extension of a totally real number field $F$ and let $p > 2$ be a rational prime. My question simply is Are the $p$-th roots of unity, ...
1
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1answer
86 views

Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. ...
5
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1answer
56 views

Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
3
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1answer
69 views

Conjecture about some group ring representations.

In this link : http://bandtechnology.com/PolySigned/ A set of numbers is described : $P(N)$. $ P(3),P(4),P(5),... $ are all (algebraicly closed) group rings. Identify $PN$ with ...
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0answers
61 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
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0answers
44 views

The Maclaurin expansion of $\prod_{k=0}^{n-1}\sin\left(\sqrt{\zeta_n^k}x \right)$.

Specifically, I'm interested in the $x^{3n}$ coefficient. The reason for my curiosity is a proof of the closed form of $\sum_{k \ge 1} \frac{1}{k^{2n}}, n \in \mathbb{N}$. Starting with the ...
0
votes
1answer
44 views

Adjoining two primitive n-th roots

Let $\omega_n$ denote a primitive $n^{th}$ root of unity. If $m$ and $n$ are positive integers with $lcm(m,n)=k$, show that $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_k)$. To start, I am aware ...
1
vote
2answers
38 views

$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$

$$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$$ where $c \in \mathbb Z_9$, $w=e^{2\pi i/9}$ and $\mathbb Z_9$ is the ring of integers modulo 9.
1
vote
1answer
44 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
3
votes
2answers
82 views

evaluating norm of sum of roots of unity

let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?