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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2answers
35 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
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1answer
32 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
79 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
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1answer
43 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 ...
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2answers
33 views

If $\omega$ is cube root of unity, form an equation whose roots are $3\omega$ and $3\omega^2$. [closed]

If $\omega$ is cube root of unity, form an equation whose roots are $3\omega$ and $3\omega^2$.
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1answer
27 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
38 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
158 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
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1answer
11 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
70 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
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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 ...
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1answer
43 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
40 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
53 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
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3answers
72 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?
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1answer
43 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
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3answers
62 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)
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1answer
28 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
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2answers
34 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
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0answers
20 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
55 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
32 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
68 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
69 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
55 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
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2answers
62 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
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1answer
48 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
64 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 ...
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votes
2answers
71 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
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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
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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
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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
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0answers
29 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
72 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
120 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
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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 ...
3
votes
1answer
80 views

A Polygon is inscribed in a circle $\Gamma$

A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove ...
0
votes
1answer
47 views

To prove: For $p$ prime, $\zeta \in \mathbb{C}*$ and $ord(\zeta) = p$, $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$

I'm trying to solve the following: Let $p$ be a prime number and $\zeta \in \mathbb{C}^* $ such that $ord(\zeta) = p$. Show that $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$ and that there is ...
3
votes
1answer
89 views

To prove: $ [K : \mathbb{Q}] = 2 \ \Longrightarrow \ \exists \zeta \text{ primitive root of unity}, \ \mathbb{Q}(\zeta) \ \supseteq \ K $

I have to show that the following statement is true: Let $K$ Be a field extending $\mathbb{Q}$ such that $[K: \mathbb{Q}] \ = \ 2$. Then there is a root of unity $\zeta$ such that $K \subseteq ...
0
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0answers
58 views

Roots of unit in a DVR

Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$. I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the ...
3
votes
0answers
41 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
1
vote
2answers
28 views

Are all roots of unity (solutions to $a^k \equiv 1$) for a prime modulo $p$, a multiple of $p-1$??

If $a$ is coprime to $p$, and $a \not \equiv 1,-1 \mod p $ then are there any solutions to $a^k \equiv 1 \mod p $ such that $0< k < p-1$? For any counterexample, it is obvious $GCD(p-1, k) \not ...
3
votes
2answers
61 views

Roots of $z^r=1,r\notin\mathbb{Q}$

If $a,b\in\mathbb{Z}$, and $\frac a b$ is in lowest terms, then $$z^{\frac a b}=1\\\implies z=\exp\left(\frac{2\pi in b}{a}\right)\forall n\in\mathbb{Z}$$ This means that $z$ has exactly $a$ distinct ...
0
votes
2answers
61 views

Find roots of polynomial

Let p be an odd prime number and $ζ =ζ_p = cos(2π/p)+isin(2π/p)$ How do you find all of the roots of the polynomial $f(x) = x^{p-1} +x^{p-2}+…+x+1$ How do you show that $p = (1-ζ)(1-ζ^2)…(1-ζ^{p-1}) ...
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votes
1answer
41 views

Finding the elements of a number C based off of 24th and 54th roots of unity

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: \begin{align*}A&=\{z\;|\;z^{24}=1\}\\ ...
1
vote
1answer
25 views

Rings and equations

Let $R$ be a commutative (non-zero) ring with identity, what are the solutions of $x^2-1=0$? Obviously, $x=\pm 1$ are solutions and if $R$ is an integral domain there aren't other solutions since ...
2
votes
1answer
44 views

Cyclic properties of multiplicative group G of all the complex $2^n$ roots of unity

Consider the multiplicative group G of all the complex $2^n$ roots of unity, $n=0,1,2,\ldots$ I am asked to verify whether $G$ is a cyclic group and whether it has a finite set of generators. The ...
1
vote
1answer
25 views

Rational number in $\mathbb{Z}[\omega]$ should be integer.

Let $\omega = \cos \frac{2\pi}{p} + i \sin \frac{2\pi}{p}$ for some prime number $p > 2$. Then how to prove that if $q \in \mathbb{Q} \cap \mathbb{Z}[\omega]$, $q$ must be integer.