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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-3
votes
1answer
21 views

Fourth roots of a certain complex number [on hold]

Find the fourth roots of $81(\cos 320^\circ + i\sin 320^\circ )$. Write the answer in trigonometric form. \begin{array} \text{a.} & 3(\cos 160^\circ + i \sin 160^\circ ); & &3(\cos ...
2
votes
0answers
51 views

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 ...
1
vote
1answer
38 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
0
votes
0answers
30 views

Linear combinations of roots of unity forming a commutative ring

How can it be shown that $$\mathbb{Z} [\zeta] = \{a + b\zeta^k \mid \zeta \text{ is a primitive $n$th root of unity; } 0 \le k \lt n ; \text{and } a,b \in \mathbb{Z} \} $$ is closed under addition? I ...
4
votes
0answers
42 views

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 ...
1
vote
0answers
21 views

What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition ...
0
votes
2answers
32 views

Number of elements in sets of roots of unity

Problem: 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 ...
0
votes
1answer
26 views

Equation with binomial coefficients

Problem: Find the roots of $6z^5+15z^4+20z^3+15z^2+6z+1 = 0$. What I found: I realized that the coefficients were the binomial coefficients of $6$. Putting these values in, you would get ...
4
votes
0answers
32 views

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 ...
1
vote
1answer
53 views

Proof Check: automorphism sends primitive root to primitive root

I was just wondering if this is a valid proof. I am assuming knowledge that if $\phi$ is an automorphism of a numeric field the $\phi$ fixes $\mathbb{Q}$. Also, if $\phi \in$ ...
1
vote
2answers
45 views

Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of ...
2
votes
1answer
45 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
0
votes
1answer
25 views

Constructibility of sum of $n$-th roots of unity

For $S \subset \{z \in \mathbb{C}: z^{11}=1\}$ we define $z_s= \sum_{s \in S}s$. Let $\zeta \in \mathbb{C}$ be a primitive $11$th root of unity. Is $z_s \in \mathcal{C}(0,1)$, for ...
5
votes
2answers
256 views

Probability a polynomial has a root which is a root of unity

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently. What is the probability that $P(x)$ has a root which is a root of unity? ...
0
votes
3answers
50 views

Root of unity: Is it true that $w_N = w_N^{(N-1)(N-1)}$ and why?

Is it true that for the $N$th root of unity $w_N = w_N^{(N-1)(N-1)}$ and why?
3
votes
3answers
74 views

$\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ Galois group

How do I calculate the degree of $\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ over $\mathbb{Q}$. Should it be 18, as $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}] = 3$, and $[\mathbb{Q}(\zeta_{9}):\mathbb{Q}] = 6$? ...
18
votes
2answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
2
votes
1answer
55 views

Why are two statements about a polynomial equivalent?

I am reading a claim that the following two statements are equivalent. One of the roots of a polynomial $v(t)$ is a $2^j$-th root of unity, for some $j$. The polynomial $v(t)$ is divisible either by ...
2
votes
0answers
63 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 ...
1
vote
0answers
40 views

How to quickly find a root of unity in a ring?

Lets say we're in a field where multiplication and addition are modded against some prime number P (so it's defined for {0,....,P-1} Lets fix a number N < P, such that a root of unity can be ...
2
votes
1answer
43 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 ...
1
vote
1answer
77 views

Minimal polynomials and degree of field extension

I have a cyclotomic field $\mathbb{Q}(\zeta_3)$, and want to know how I can find a minimal polynomial of $\zeta_{10}$, and $\zeta_{12}$. I have determined that both the polynomials should be of ...
0
votes
0answers
21 views

$b_k = \sum\limits_{j=0}^4 j\omega^{-kj}$, for $0\le k\le4$ $\Rightarrow$ $\sum\limits_{k=0}^4 b_k\omega^k$ =?

Let $\omega$ denote a complex fifth root of unity. Define $b_k = \sum\limits_{j=0}^4 j\omega^{-kj}$, for $0\le k\le4$. Then find the value of $\sum\limits_{k=0}^4 b_k\omega^k$.
3
votes
3answers
79 views

Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
0
votes
1answer
36 views

sum of powers of principal tenth root of unity

Set $w=\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}$. I have to calculate: $$1 + \sum_1^9 w^n$$ I have calculated that the answer is 0. However, I am supposed to arrive at this conclusion without ...
1
vote
1answer
58 views

Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
1
vote
1answer
56 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 ...
4
votes
1answer
102 views

Is the sum of roots of unity always a real multiple of a root of unity?

I can see this is true for the sum of two roots of unity with some basic trigonometry (the resulting argument is the half the sum of the original arguments, and so must also be a rational multiple of ...
6
votes
1answer
104 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 ...
1
vote
0answers
23 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 ...
0
votes
0answers
35 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 ...
0
votes
0answers
23 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 ...
1
vote
0answers
18 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
vote
1answer
59 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 ...
0
votes
1answer
159 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? ...
2
votes
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 ...
5
votes
3answers
88 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 ...
0
votes
3answers
134 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
45 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 ...
0
votes
0answers
42 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) ...
1
vote
4answers
40 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
votes
3answers
93 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
61 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
votes
1answer
65 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
votes
1answer
55 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 ...
1
vote
0answers
60 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 ...
0
votes
1answer
43 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 ...
1
vote
1answer
38 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 ...
0
votes
4answers
62 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
vote
1answer
51 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 ...