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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4
votes
2answers
79 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
1
vote
0answers
18 views

$p$-divisibility and $q$-th roots of unity

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in ...
2
votes
3answers
65 views

Product of the difference of $n$th roots of $-1$ [closed]

If $w_1,w_2,\ldots,w_n$ are the $n^{\text{th}}$ roots of $-1$, then how can we prove that by mathematical induction $$(w_2-w_1 )(w_3-w_1 )\cdots(w_n-w_1 )=\frac n{w_1}?$$
1
vote
1answer
45 views

$p^a\mid f(v) \implies p^a\mid f(w)$ in $\mathbb Z[w]$

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in ...
2
votes
0answers
54 views

Galois group over $\mathbb{Q}$ [closed]

Let $$\begin{align*} K&=\mathbb{Q}(\{\text{all $2^n$-th roots of unity for $n\in\mathbb{N}$}\})\\ L&=\mathbb{Q}(\{\text{all $n$-th roots of unity for $n\in\mathbb{N}$}\}) \end{align*}$$ What ...
2
votes
4answers
93 views

What is the cardinality of the set of roots of unity?

Consider the geometric interpretation of "roots of unity": My intuition says that you can place arbitrarily many equidistant points on the unit circle and catch every point that lies on it. ...
3
votes
2answers
28 views

Let $w$ be a primitve third root of unity. Find the units of $A=\{a+bw, a,b \in \mathbb{Z}\}$

What I have so far: if $x \in \mathbb{C}$, then $N(x)=\bar{x}x$ is multiplicative ($N(xy) = N(x)N(y)$). So $N$ restricted to $A$ is also multiplicative. if $a+bw \in A$, then it's easy to see that ...
1
vote
1answer
19 views

Does this matrix involving roots of unity has a particular name?

Do the matrices of the form $\left(\begin{matrix} \frac{\xi + \xi^{-1}}{2} & \frac{\xi - \xi^{-1}}{2} \\ \frac{\xi - \xi^{-1}}{2} & \frac{\xi + \xi^{-1}}{2} \end{matrix}\right)$ where $\xi$ is ...
1
vote
1answer
46 views

Where's the flaw in this application of De Moivre's fomula to find n-th roots?

To find the $n^\text{th}$ roots of a complex number, we can first express it in polar form (I'm assuming $r=1$ for brevity; it doesn't matter for my question): \begin{align} e^{i\theta} &= ...
0
votes
5answers
29 views

Roots of Unity: second largest value and absolute value

Consider the $n$th roots of unity $e^{2 \pi i k/n}$ for fixed integer $n \geq 2$ and $0 \leq k < n$. Now I am interested in the second largest value (in absolute value) of the values ...
0
votes
1answer
24 views

A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity.

Could someone explain to me the following sentence? "A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity." Does this mean that for each element $x$ of a field of ...
0
votes
0answers
23 views

$(ab^{n+1})^{1/n}$ where $a,b \in \mathbb{C}$

Let $a,b \in \mathbb{C}$ and $n \in \mathbb{N}$. We can present $a$ and $b$ in polar form as \begin{equation} a = r_a \mathrm{e}^{i \theta_a} \quad \textrm{and} \quad b = r_b \mathrm{e}^{i \theta_b}, ...
2
votes
1answer
262 views

Find roots of unity

Find the roots of $6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 = 0.$ I know how to do this without the coefficients, but I do not know what to do in this problem. Thanks
1
vote
2answers
73 views

Complex numbers - roots of unity

Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 ...
2
votes
1answer
51 views

Primitive roots of unity [duplicate]

Let $R$ be the set of primitive $42^{\text{nd}}$ roots of unity, and let $S$ be the set of primitive $70^{\text{th}}$ roots of unity. How many elements do $R$ and $S$ have in common? How would you ...
0
votes
5answers
87 views

Compute $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$ where $\omega^3 = 1$

If $\omega^3 = 1$ and $\omega \neq 1$, then compute $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$ I'm pretty lost, I don't really know where to start. Thanks
1
vote
1answer
46 views

Series involving complex roots

$$ \frac{1}{2-a_1} + \frac{1}{2-a_2} + \dots + \frac{1}{2-a_{n-1}} = \frac{(n-2)2^{n-1}+1}{2^n - 1} $$ 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 ...
0
votes
1answer
31 views

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

Find a $7^{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}$, so that $$ p(14^i) = i\ (mod\ 41)\ \forall i = 0,1,\ldots,7. $$ $3$ is the $8^{th}$ primitive root of unity and $3 * 14 = 8 * 36 = 1$ ...
4
votes
2answers
41 views

Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.

I am trying to solve the following exercise in Koblitz's "$p$-adic Numbers, $p$-adic analysis, and Zeta-Functions". Let $p$ be a prime. Let $a$ be a primitive $p^n$-th root of unity in ...
2
votes
2answers
161 views

A unit of seventh cyclotomic field

I have troubles with the following problem about units. Show that $1+\zeta $, $1+\zeta+\zeta^2$ are units in the field $\mathbb{Q[\zeta]}$, where $\zeta$ is a seventh primitive root of unit ...
0
votes
1answer
17 views

Primitive roots of unity proof verification

"Let $C_n(x)$ be the polynomial such that the roots of $C_n(x)=0$ are the primitive $n^{th}$ roots of unity. Prove that there are no positive integers $q,r,s$ for which $C_q(x)=C_r(x)C_s(x)$." My ...
4
votes
1answer
73 views

Can $\sin(\pi/25)$ be expressed in radicals, revisited

This was inspired by this post. Let, $$q = e^{2\pi\, i/m}$$ D. Speyer's answer can be generalized as, $$\sin\Big(\frac{\pi}{m^2}\Big) = \frac{i}{2}\Big(-q^{1/(2m)}+q^{-1/(2m)} \Big)\tag1$$ while ...
1
vote
1answer
34 views

Square root of $-1$ over a finite field [duplicate]

It is known that the equation $x^2 \equiv -1 \pmod{p}$, where $p$ is an odd prime number, has a solution iff $p = 4k +1$ for some natural $k$. Does it exist a similar characterization for a general ...
-1
votes
1answer
48 views

Finding a primitive fifth root of unity modulo $81$ using a specific method.

I want to find a fifth root of unity modulo $81$ using a suggested method from the book (I can't come up with any other good method anyway). It is given that $x^4+x+2 \in \mathbb{F}_3[x]$ is ...
3
votes
1answer
41 views

Every unit in $\mathcal O_K$ is equal to a power of $\zeta$ times a real unit in $\mathcal O_K$

Every unit in $\mathcal O_K$ is equal to a power of $\zeta$ times a real unit in $\mathcal O_K$, with $\zeta:=e^{2\pi\sqrt{-1}/p}$ and $K:=\mathbb Q(\zeta)$ The proof is below, but I don't ...
3
votes
3answers
54 views

Roots of unity are distincts

For every $n\in\Bbb N$ and $$z_{k}:= \cos(2\pi k /n)+i\sin(2\pi k /n), \qquad k = 0,\ldots,n-1$$ we have $z_k^n=1$. How to show, in a simple way, that $z_k\neq z_l$ for every $k\neq l$? By ...
1
vote
3answers
74 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
0
votes
1answer
45 views

A problem in understanding principal root in the complex plane.

We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part ...
1
vote
3answers
69 views

Cubic root of unity

Is there anyway to solve this without substituting with the values? Prove that: $$\frac{1+10w^2}{1-2w} + \frac{2+17w}{2+3w} = 6$$. (Where $w$ & $w^2$ are the cubic roots of unity)
0
votes
1answer
29 views

Showing $\zeta_5 \notin \mathbb{Q}(\zeta_7)$

I was assigned this problem as homework, and got it wrong. I have not gotten a chance to ask the teacher about the solution. Can someone tell me why I am wrong, and how to do this correctly? Let ...
0
votes
2answers
37 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
0
votes
0answers
23 views

Roots of unity: Bounds on Eigenvalues of circulant matrix

Can you tell me a bound for $$\left|\sum_{j=0}^{k^n-1}c_j e^{2\pi i j \frac{m}{k^n}}\right|, \quad m \in \{0, \dotsc, k^n-1\}$$ the absolute values of the eigenvalues of a circulant matrix with ...
2
votes
1answer
59 views

Find the minimum polynomial of a sum of roots of unity.

Let $ \omega $ be an 11-th primitive root of 1 over $ \Bbb Q $ Let $ \beta = \omega + \omega^9 $ Find $ [ \Bbb Q ( \beta) : \Bbb Q ) ] $ and Find the minimum polynomail of $\beta$. I asked a ...
0
votes
2answers
33 views

solving the limit $\lim_{n\to \infty}\sum_{k=1}^n|e^{(2πik)/n}-e^{(2πi(k-1))/n}$|

$$\lim_{n\to \infty}\sum_{k=1}^n\left|e^{(2πik)/n}-e^{(2πi(k-1))/n}\right|$$ i can solve it geometrically. but is there any way to solve it using Euler's formula ?, the answer will be one of these ...
0
votes
3answers
76 views

Roots of Unity, Precalculus

(a) Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 ...
0
votes
3answers
87 views

Precalculus Roots of Unity

Let $A$ be the set of all complex numbers $z$ such that $z^{24}=1$ and let $B$ be the set of all complex numbers $w$ with $w^{54}=1.$ That is \begin{align*} A&=\{z\;|\;z^{24}=1\}\\ ...
2
votes
2answers
61 views

Roots of Unity for Precalculus

(a): What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity? (b) What is the smallest positive integer $n$ such that all the roots ...
2
votes
2answers
43 views

If $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then $R^{\times}=\mathbb Z\big/6\mathbb Z$

How to show that if $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then the group of units $R^{\times}=\mathbb Z\big/6\mathbb Z$ Now since $-3\equiv1\mod 4$ the ring ...
1
vote
2answers
80 views

How many distinct elements are there in $C=\{zw\mid z∈A$,$w∈B\}, z^{24}=1$ and $w^{54}=1$.

Let $A$ be a set of all complex numbers $z$ such that $z^{24}=1$ and let $B$ be the set of all complex numbers $w$ such that $w^{54}=1$. That is: $A$={$z$|$z^{24}=1$} and $B$={$w$|$w^{54}=1$} ...
1
vote
1answer
52 views

number of roots of unity which satisfy a given polynomial

Let $A$ be a matrix over $\mathbb{R}$ and $p_A(x)$ its characteristic polynomial. Is there an easy way to find out how many of the roots of $p_A(x)$ are roots of unity? Fixing a positive integer $k$, ...
2
votes
1answer
65 views

Roots of unity of an odd degree number field

I want to show that a number field of odd degree contains only $2$ roots of unity. The only information I really have regarding this that I think is relevant is that the group of units ...
-1
votes
2answers
75 views

Sum of roots of unity a root of unity

Question is to check if : $a_1,a_2$ are $n^{\rm{th}}$ roots of unity and $|a_1+a_2|=1$ imply $a_1+a_2$ is a root of unity... A more general question is asked Sums of roots of unity but they are ...
0
votes
0answers
86 views

Carmichael function and primitive roots of unity

I have been reading about the Carmichael function recently and I would like to ask about some elementary implication of its properties as I haven't found it stated explicitly. If I understand it ...
0
votes
0answers
34 views

Carmichael numbers and primitive roots of unity

Let $n$ be a Carmichael number. Is it possible for an integer ring $\mathbb{Z}_n$ to contain primitive $(n-1)^{th}$ roots of unity? Or do only only primitive roots of unity of degree $\quad k < ...
1
vote
1answer
32 views

Prove: there is a unique pair of integer roots of unity which differ in real part by $1$.

I saw the following lemma somewhere, and I hope I did not misread it: If $z_1$ and $z_2$ are $n$th and $m$th roots of unity respectively ($n,m$ positive integers possible equal), and the real part of ...
0
votes
0answers
31 views

Fast Fourier Transform: How is the roots of unity matrix divided?

For an example for input size N=8, how is the roots of unity matrix divided for a divide and conquer approach? My understanding is that it's divided into four quadrants, Ma with J&K evens; Mb ...
0
votes
3answers
75 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
0
votes
2answers
53 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
4
votes
5answers
121 views

Why is $\sqrt [n] 1$ not an expression “in radicals” of a root of unity?

In Edwards' Galois Theory, in the chapter on Cyclotomic polynomials, the author devotes a lot of effort to proving that prime order primitive roots of unity can be expressed "by radicals", and gives ...
0
votes
1answer
33 views

for $p$ given, $\zeta_p$ a primitive root of unity, fow which $d\in \mathbb{Z}$ does $\zeta_p \in \mathbb{Q}(\sqrt{d})$?

Here is a question that I am trying to answer: Let $p$ be a prime greater than $2$. For which $d \in \mathbb{Z}$ contains $\mathbb{Q}(\sqrt{d})$ a primitive root of power $p$? What I did If ...