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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1
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4answers
107 views

Find all $ z \in {\mathbb C} $ such that $z^{12}=1 $ and $ 1+z+z^2+z^3+z^4+z^5 \in {\mathbb R} $

So, my first thought. If $z^{12} = 1, z \in $ is a twelfth root of unity. Knowing this, I can write $ z = e^{i {2k \pi} \over {12}} $, with k $\in \{0,1,2,3,4,5,6,7,8,9,10,11\} $. Then if I just ...
1
vote
1answer
35 views

If $k \in \mathbb{N}, n={2^k} $. Probe that $w$ is a primitive nth-root of unity $\iff$ $w$ is a root of $ P_k = x^{2^{k-1}} + 1 $

Going to the right is understandable. $w$ is a $2^k $-th primitive root of unity $\implies$ $w$ is a root of $P_k = x^{2^{k-1}} $ $ w \in G_{2^k} \implies w^{2^{k}} = 1$ But then $w$ is a ...
1
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5answers
60 views

$ p \in Q[x] $ has as a root a fifth primitive root of unity, then every fifth primitive root of unity is a root of $p$.

I'm extremely stuck. Can't figure it. The conjugate is easy: let $w$ be a primitive root of unity, then $w^{-1}$ will also be a root, that's easy. But I'm missing $w^2$ and $w^3$. Why would they be ...
0
votes
0answers
31 views

Complex roots for integer polynoms $p(z)=0$ with restriction $| z|=1$

There have been may questions about (integer) polynomials of sin and cos. There have been nearly as many ingenious answers using trigonomic-magic (converting sin, cos, tan into each other), but I (not ...
1
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1answer
39 views

If $X$ has character $\chi$ and degree $d$. Prove that $g \in N$ if and only if $\chi(g) = d$ . Hint: Show that $\chi(g)$ is a sum of roots of unity.

Let $X$ be a matrix representation. And $N ={\{g \in G: X(g) = I}\}$. If $X$ has character $\chi$ and degree $d$. Prove that $g \in N$ if and only if $\chi(g) = d$ . Hint: Show that $\chi(g)$ is a ...
1
vote
3answers
86 views

Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Background This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties,...
0
votes
2answers
66 views

Finding fifth and tenth roots of unity in rectangular form.

Exercise: Find the fifth and tenth roots of unity in algebraic form. This is an early exercise in Ahlfors Complex Analysis. What I have tried so far: For the fifth roots I have tried reducing the ...
5
votes
1answer
115 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find $\sqrt{...
3
votes
1answer
70 views

Show that the elements of the form $1+\zeta + \zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$

Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$. I know ...
0
votes
3answers
45 views

Linear Combination of Roots of Unity

Let $\omega_n$ be a primitive $n$the root of unity and $\lambda_k$ be natural numbers. Does $\sum_{k=1}^{n} \lambda_k w_n^k =0$ imply $\lambda_1 = \lambda_2 = ... = \lambda_n $? I am aware of ...
0
votes
1answer
26 views

Why is the sum of the first $k$ powers of a $k$-th primitive root $\varphi_k$ of $1$ always $0$?

Let $\varphi_k\in\mathbb{C}$ be a primitive root of $1$. It turns out, that $$\varphi_k^1+\ldots+\varphi_k^k=0\text{ .}$$ If I draw the roots for some fixed $k$, I can see that this seems evident. For ...
5
votes
2answers
71 views

Find complex roots of $\frac{2i}{1+i}$

Find 6th roots of $$\frac{2i}{1+i}$$ $$\frac{2i}{1+i}=\frac{2e^{i\pi/2}}{\sqrt 2 e^{i \pi/4}}=\sqrt 2 e^{i \pi/4}$$ Now if I set $z^{1/6}=\sqrt 2 e^{i \pi/4}$ and knowing the fact that the roots ...
0
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1answer
32 views

What is the intersection of $\mathbb Z$ with the ideal generated by $1-\zeta_n$?

For example, 1-(-1) is in the ideal <2>, whenever $n$ is even. Suppose $R=\Bbb Z[\zeta_n]$, and (by the above), that $n$ is odd. We know 1+$\zeta_n$ can be multiplied by $1+\zeta_n^2+\zeta_n^4+\...
4
votes
2answers
80 views

Why do I get an imaginary result for the cube root of a negative number?

I have a function that includes the phrase $(-x)^{1/3}$. It seems like this should always evaluate to $-(x^{1/3})$. For example, $-1 \cdot -1 \cdot -1 = -1$, so it seems that $(-1)^{1/3}$ should equal ...
0
votes
2answers
33 views

Why in a field of characteristic $p$, $\zeta_p \sqrt[p]t$ is not a root of $X^p-t\in \mathbb F_p(t)[X]$

I know that in a field such that $Car(K)=p$ is prime, the $X^p-t\in \mathbb F_p(t)[X]$ has a unique root (I know how to prove it and thus, it's not the question). But in the usual logic, $X^p=t\iff X=\...
3
votes
1answer
59 views

What is the sum of ALL of the nth powers of the qth roots of unity?

I am reading a Wikipedia article on Ramanujan's sum. The article states that: It follows from the identity $x^q − 1 = (x − 1)(x^{q−1} + x^{q−2} + ... + x + 1)$ that the sum of the nth powers of All ...
1
vote
0answers
29 views

How can I choose $\frak p\unlhd\cal O$ prime so $u\in\cal O^\times$ becomes a $n$-th power (mod $\frak p$)?

$k$ is an algebraic number field, and $\cal O$ is the ring of integers, $\cal O^\times$ is the set of invertible elements of $\cal O$. Suppose $u\in\cal O^\times$ is not a $n$-th power. How can I ...
3
votes
4answers
85 views

All Solutions for $(-256)^{\frac{1}{4}}$ and $1^{\frac{1}{5}}$ Imaginary Roots?

This is a question about the imaginary roots of the two equations $$ (-256)^{\frac{1}{4}} \qquad\text{and}\qquad 1^{\frac{1}{5}}. $$ For the first one I've worked out that 2 of the solutions are $4\...
-2
votes
2answers
139 views

What is $1^{1/n}$ for $n=\infty$ or for irrational $n$? [closed]

The $n$th root of $1$ produces a $n$ sided regular polygon centered around $0+0i$ with a vertex at $1+0i$. However, if we have $n=\infty$, then we should get an infinite sided polygon? Or more ...
0
votes
1answer
30 views

On multiplying symmetric matrices by diagonal matrices with roots of unity

Given two symmetric non-zero and non-identity matrices $A,B\in\Bbb C^{n\times n}$ of same rank supposing there exists a non-identity diagonal matrix $D\in\Bbb C^{n\times n}$ containing only roots of ...
0
votes
1answer
37 views

nth root of unity in a cyclic group $\mathbb{Z}_p^*$

Is there a specific set of steps that should be taken in order to the $n$-th root of unity in a cyclic group. To be more specific, I am trying to find the $8$th root of unity for $\mathbb{Z}_{17}^*$. ...
3
votes
4answers
120 views

Prove that $\cos\left(\frac{2\pi}{n}\right)+\cos\left(\frac{4\pi}{n}\right)+\ldots+\cos\left(\frac{2(n-1)\pi}{n}\right)=-1$

May you help on how to start, or where to look for the following question? By using the $n$-th roots of the unity, show that: $\cos\left(\frac{2\pi}{n}\right)+\cos\left(\frac{4\pi}{n}\right)+\ldots+...
3
votes
2answers
127 views

Adding primitive $n^\text{th}$ roots of unity, where $n$ is not square-free.

I want to show that, for $n$ not square-free, $$\sum\limits_{\substack{1\leq k \leq n\\ \gcd(k,n)=1}} \xi _n^k=0,$$ where $\xi_n$ is a (fixed) primitive $n^\text{th}$ root of unity (in $\mathbb C$)...
2
votes
4answers
86 views

What are the 4th degree roots of $1$?

So my question is the fourth root of $1 = i$? Where $i^4 = 1$? or if it is still $1$ nonetheless?
0
votes
1answer
32 views

Complex Numbers Midpoint of Roots of Unity

A = $\sqrt{2}e^{i(\frac{7\pi}{12})}$ B = $\sqrt{2}e^{i(\frac{11\pi}{12})}$ Express the midpoint M of AB in the form $a + bi$ (a,b in simplified surd form) I know M = (A+B)/2 but I cant find A+B in ...
2
votes
1answer
29 views

Let $\xi_{19} \in G_{19}$ be a primitive root of unity. Find $\Re\sum_{k=1}^9\xi_{19}^{k^2}$.

Question Let $\xi_{19} \in G_{19}$ be a primitive root of unity. Find $\Re\sum_{k=1}^9\xi_{19}^{k^2}$. Attempt I'm having doubts about how I'm solving this exercise, this is what I did: $$ \Re\...
3
votes
0answers
82 views

Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
4
votes
1answer
47 views

Find the value of:$\frac{|1-z_1||1-z_2||1-z_3|\ldots|1-z_9|}{10}$where $z_k$ is the 10th root of unity

$$z_k=\cos\left( \frac{2k\pi}{10}\right)+i\sin\left(\frac{2k\pi}{10}\right);k\in \{1,2,3,\ldots,9\}$$ then find the value of:$$\frac{|1-z_1||1-z_2||1-z_3|\ldots|1-z_9|}{10}$$ Answer: 1 My attempt:...
1
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0answers
22 views

If a, b, and c are real, and w is a non real cube root of unity…

If a, b, c, and d are real numbers and w is a non-real cube root of unity, and if 1/(a + w) + 1/(b + w) + 1/(c + w) + 1/(d + w) = 2/w Prove that 1/(a + w^2) + 1/(b + w^2) + 1/(c + w^2) + 1/(d + w^2)...
2
votes
1answer
70 views

Can we write $\sqrt2$ and $2$ by using $16$. primitive roots of unity

Edited version: Are there integers $a_i$ such that $$\sqrt{2^k}=\sum\limits_{k=0}^3 a_{2k+1}e^{(2k+1)i\pi/8}$$ for some integer $k>0$? Previous version: Are there integers $a_i,b_i$ such that $\...
3
votes
0answers
36 views

Proving unit of quartic number field is fundamental

Let $K = \mathbb Q(\alpha)$, for $\alpha$ a root of $a^4 + 4 \alpha^2 + 2 = 0$. I want to prove the group of units $\mathcal O_K^*$ equals $\langle -1, \alpha^2 + 1\rangle$. I've found the ring of ...
1
vote
2answers
38 views

Trace of finite-order matrix over $\Bbb C$.

Suppose that $A\in M_n(\Bbb C)$ is such that $A^m=Id$ for some $m\in > \Bbb N$, is it true that if $Tr(A)\in \Bbb R$ then it must be an integer? Here, $Tr(A)$ denotes the sum of the elements in the ...
5
votes
1answer
154 views

Trace of Product of Powers of $A$ and $A^\ast$

Let $n$ be odd, $\displaystyle v=1,...,\frac{n-1}{2}$ and $\displaystyle \zeta=e^{2\pi i/n}$. Define the following matrices: $$A(0,v)=\left(\begin{array}{cc}1+\zeta^{-v} & \zeta^v+\zeta^{2v}\\ \...
2
votes
2answers
34 views

Prove that function $g(n)=\zeta^{5n}$ is surjective from $\mathbb Z$ to the set of roots of unity

I have this math problem, that I'm kind of stuck on. $\mu_{102} = \{ z \in \mathbb{C}: z^{102} = 1\}$ Let $\zeta = e^{\frac{2 \pi i}{102}}.$ Define $g : \mathbb{Z} \to \mu_{102}$ with the ...
1
vote
3answers
55 views

$o(z) = 28$, what is $o(z^{16})$?

I have this math question that I am kind of stuck on. Suppose that $o(z) = 28$, what is the order of the element $z^{16}$? From this information I know that $o(z) = 28$ means that $z^{28} = 1$......
1
vote
1answer
30 views

Simplifying Exponentials (Fourier)

I am having trouble simplifying the following expression: $$ \frac{1}{7}\left(1+e^{-jk\frac{2\pi}{7}}+e^{-jk\frac{4\pi}{7}}+e^{-jk\frac{6\pi}{7}}+e^{-jk\frac{8\pi}{7}}\right) $$ I need to get $$ \...
2
votes
2answers
36 views

Prove root of unity and order

I have this math problem: i) Suppose that $a$ and $b$ are roots of unity. Suppose that $o(a)=5$ and $o(b)=7$. Prove that $o(ab)=35$. ii) Give an example such that $a$ and $b$ are roots ...
0
votes
3answers
31 views

Show root of unity and order

I have this math problem: Set $$z=\frac{1}{2}-\frac{\sqrt{3}}{2} i$$ Show that $z$ is a root of unity, find its order, and express $z^{100}$ in the form $a+bi$. I'm not 100% sure how to do ...
-1
votes
1answer
30 views

Show root of unity summation

I have this math problem Let $w$ be a root of unity with $o(w)=n$, with $n > 1$. Show that $$1 + w + w^2 + \cdots + w^{n-1} = 0$$ I'm not entirely sure how to start this problem. Would I ...
2
votes
2answers
55 views

Of sum of cosines and the $7$th roots of unity

In my solution here, it was shown that $$\omega+\omega^2+\omega^4=-\frac 12\pm\frac{\sqrt7}2\qquad\qquad (\omega=e^{i2\pi/7})$$ from which we know that $$\sin \frac{2\pi}7+\sin \frac{4\pi}7-\sin \...
18
votes
7answers
2k views

Find all five solutions of the equation $z^5+z^4+z^3+z^2+z+1 = 0$

$z^5+z^4+z^3+z^2+z+1 = 0$ I can't figure this out can someone offer any suggestions? Factoring it into $(z+1)(z^4+z^2+1)$ didn't do anything but show -1 is one solution. I solved for all roots of $...
0
votes
1answer
48 views

Approximation of $\cos\frac{2\pi}{5}$ and $\sin\frac{2\pi}{5}$ using solutions of $z^5-1=0,z\in\mathbb{C}$

One one hand, $\cos\frac{2\pi}{5}$ and $\sin\frac{2\pi}{5}$ are values of trigonometric functions; on the other hand, $\cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}$ is a root of algebraic equation $z^5-1=...
1
vote
0answers
32 views

Is there always a zero of this polynomial that is not a root of unity?

Consider the polynomial $1-z(1+z)^a$ with $a$ a positive integer. Is there always a complex zero of this polynomial that is not a root of unity? I tried to prove it by induction or by contradiction ...
-1
votes
4answers
87 views

Why does the De Moivre formula work?

In order to find the nth roots of a complex number you need to use this formula. https://en.wikipedia.org/wiki/De_Moivre%27s_formula#Roots_of_complex_numbers For me I'm trying to understand where ...
6
votes
4answers
129 views

Algebraic values of sine at sevenths of the circle

At the end of a calculation it turned out that I wanted to know the value of $$\sin(2\pi/7) + \sin(4\pi/7) - \sin(6\pi/7).$$ Since I knew the answer I was supposed to get, I was able to work out that ...
0
votes
1answer
132 views

A proof of root of unity

Let $\omega$ be the root of unity $e^{2\pi i/90}$, prove that $$ \prod_{n=1}^{45}\sin(2n^\circ)=\sum_{n=1}^{45}\frac{\omega^n-1}{2i\omega^{n/2}}. $$
0
votes
1answer
28 views

Finding one of the roots of an equation

I am trying to one of the roots of the following equation $$z^5 = -16 + (16\sqrt 3)i$$ which is $$z = 2e^{\frac{(6k+2)\pi}{15}i}$$ However, I have trouble getting that root. Here is what I have done: $...
1
vote
2answers
74 views

Finding the Roots of Unity

I have the following equation, $$z^5 = -16 + (16\sqrt 3)i$$I am asked to write down the 5th roots of unity and find all the roots for the above equation expressing each root in the form $re^{i\theta}$....
0
votes
1answer
81 views

How to solve a cube root of unity (complex numbers)?

I have to do this: Find the value of $\sqrt[3]{\dfrac{1-i}{1+i}}$ in binomial and polar form. I have arrived to this point: $$z=\sqrt[3]{\dfrac{1-i}{1+i}}=\sqrt[3]{\dfrac{1-i}{1+i}\cdot\dfrac{1-...
0
votes
1answer
123 views

Find the real and imaginary parts of $z^{22}=\sqrt{3} - i$

I know how to find the roots of this answer, but I believe this question is asking for something different as I don't believe I am expected to write out all 22 roots. I am not looking for the answer ...