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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2
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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
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1answer
37 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. ...
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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
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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
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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
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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
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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
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0answers
31 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
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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
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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
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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 ...
0
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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?
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? ...
3
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3answers
73 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 $[\mathbb{Q}(\zeta_{9}):\mathbb{Q}] = 6$? ...
2
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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 ...
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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
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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
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
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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
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1answer
35 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
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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
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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 ...
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 ...
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 ...
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 ...
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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
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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 ...
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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
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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
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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 ...
5
votes
3answers
87 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
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) ...
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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
92 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
60 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
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 ...
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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 ...
1
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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 ...
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
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 ...
3
votes
1answer
122 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 ...
0
votes
1answer
35 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
votes
1answer
59 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 ...