numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

learn more… | top users | synonyms

2
votes
4answers
85 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
23 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: $$ ...
3
votes
0answers
67 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
42 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 ...
1
vote
0answers
19 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 + ...
2
votes
1answer
68 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
33 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
34 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 ...
5
votes
1answer
149 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}\\ ...
1
vote
2answers
33 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
54 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
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 ...
1
vote
2answers
49 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 ...
17
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
45 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 ...
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
86 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
126 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
67 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 ...
0
votes
1answer
78 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: ...
0
votes
1answer
106 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 ...
2
votes
0answers
38 views

How many tubes can you balance in a centrifuge?

I recently learned that if you have a centrifuge whose number of holes $n$ is divisible by $6$, then you can balance any number of tubes except for $1$ and $n-1$. If $k$, the number of tubes you want ...
1
vote
3answers
65 views

Show that $2\cos\frac\pi5\cdot\sin\frac\pi{10}=1$

We have to prove this identity : $$2\cos\frac\pi5\cdot\sin\frac\pi{10}=1$$ The book's hint is we somehow find out that $\displaystyle1+2\left(\cos\frac\pi5-\cos\frac{2\pi}5\right)$ equals something ...
4
votes
2answers
200 views

Is $\sqrt[3]{-1}=-1$?

I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction. Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: ...
1
vote
2answers
49 views

multiple sets of complex roots of a number?

I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching. So today I discovered a weird problem: If we take this equation: $$x^2=1=e^{(0i)}$$ ...
0
votes
1answer
30 views

A theorem involving the special root of an equation

Theorem: If $a$ is a special root of the equation $x^{n}-1=0$, then $a^{p}$ is also a special root of it (where $p$ is prime to $n$). I have done a proof of this theorem. Can you please tell if ...
1
vote
1answer
57 views

computing the cubed root of a complex number…

I do know how to calculate the cubed root of a complex number....like if I'm given that $x^3=p$, where $p$ is a complex number, then $$x= r^{1/3}\left(\cos\left(\frac{2k\pi+m}{3}\right) + i\sin ...
2
votes
3answers
48 views

$\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$

Show that $\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$, where $z_0,...,z_{n-1}$ are the $n$-th roots of unity. For $k=n$ it ...
4
votes
2answers
93 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
26 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
79 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
48 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
56 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
140 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
33 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
27 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
60 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
58 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
31 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
29 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
315 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
125 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
112 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
153 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