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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3
votes
2answers
73 views

Determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity

I need to determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity. At first, I expressed this number as ${\dfrac{3}{5}+\dfrac{4}{5}i}$. Then I tried to use a formula for $\sin{nx}$, where x = ...
0
votes
1answer
58 views

Find all the roots of this complex equation

Let $C$ be the set of complex numbers and $j$ the imaginary unit. Find all the roots(in $z$ $\in$ $C$) of the following equation: $$ 2z^7 + 6z^4 = z^3e^{-j{\frac π7}} + 3e^{-j{\frac π7}} $$ ...
4
votes
1answer
80 views

$\sum_{\zeta^p=1}(\zeta-1)^n$

Given $n\geq0$ let $$ z_n=\sum_{\zeta^p=1}(\zeta-1)^n $$ where $p$ is an odd prime number (summation extended to all $p$-th roots of 1). It is clear that: $z_n\in\Bbb Z$ (it's a Galois invariant sum ...
2
votes
0answers
62 views

Automorphism(Galois groups) and galois theory

I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is: Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of ...
3
votes
1answer
48 views

Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
1
vote
0answers
34 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
0
votes
1answer
43 views

Approximating a power of a root of unity to within $\delta$

I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the ...
0
votes
0answers
19 views

Quantum Fourier transform $F_N^2$

What is the square of the quantum Fourier transform? I get $1$ for the first entry in the matrix and $0$ for all other entries.
4
votes
2answers
77 views

Nth root of Unity

Hi all I am in higher level mathematics and I am taking the IB. We started doing problems associated with nth root of unity. I understand how to find the roots of for example: $$Z^3 - 1 =0$$ and ...
3
votes
4answers
98 views

Norm of an element in cyclotomic extension (Exercises VI.19 Lang's Algebra)

Let $\zeta$ be a primitive $n^{\rm{th}}$ root of unity. Let $K=\mathbb{Q}(\zeta)$. If $n=p^r (r\geq 1)$ is a prime power, show that $N_{K/F}(1-\zeta)=p$ If $n$ is divisible by at least two distinct ...
0
votes
2answers
52 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since ...
0
votes
3answers
23 views

Are conditions equaivalent that they are roots of unity?

If I have conditions that $|z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$ Is it suffice to state they are roots of unity ?
2
votes
3answers
204 views

Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...
-1
votes
2answers
57 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...
1
vote
3answers
25 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
1
vote
1answer
78 views

Product of roots of unity

Does somebody have a nice proof of the following? $$\prod_{m=1}^{n-1} \frac{e^{2\pi i k m/n} - 1}{e^{2 \pi i m / n} - 1} = \begin{cases} 1 & \text{ if $\gcd(k, n) = 1$} \\ 0 & ...
8
votes
2answers
126 views

What is the value of $\sum_{m=1}^{19} \frac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$ with $\zeta=e^{2\pi i/19}$?

Given that $\zeta=e^{2\pi i/19}$, how to find the value of $$S=\sum_{m=1}^{19} \dfrac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$$? All I could think of was to somehow factorize the denominator and apply ...
1
vote
1answer
42 views

Question Primitive Roots of Unity

I am learning about primitive $n$-th roots of unity. I came across this statements while reading and was wondering why these were true: If $z$ is a primitive $n$-th root of unity and $n$ is even, ...
3
votes
5answers
76 views

How to solve $z^6+i=0$

I'm trying to solve $z^6+i=0$. I would have say that it's equivalent to $$z^6=-i\iff |z|^6e^{i6\arg(z)}=e^{i\frac{3\pi}{2}}\iff|z|^6=e^{i\left(\frac{3\pi}{2}-6\arg(z)\right)}$$ But I'm not able to ...
1
vote
0answers
58 views

$p$-adic $n$-th root of unity and $\exp(2\pi i /n)$

Let $n\geq 3$, and let $p$ be a prime number $\equiv 1 $ mod $n$. In complex numbers, we can write a primitive $n$-th root of unity as $\exp(2\pi i/n)$. Also, by Hensel's lemma, we see that $n$-th ...
2
votes
2answers
46 views

When is a sum of consecutive roots of unity an integer

Let $\xi \neq 1$ be an $n$th root of unity. When is a sum of the form $$ 1+\xi+\xi^2+\ldots+\xi^r, \quad 1 \leq r \leq n-1, $$ an integer? What are the possible integers? I suspect that the answers ...
3
votes
4answers
123 views

$z^{10}+\frac{1}{z^{10}}=?$

$z$ is a complex number and $z^2+z+1=0$. $$z^{10}+\frac{1}{z^{10}}=?$$ For the solution: the roots of $z^2+z+1$ are: $z_1=-\frac12+\frac{\sqrt3}{2}i$ and $z_2=-\frac12-\frac{\sqrt3}{2}i$ ...
0
votes
1answer
46 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
3
votes
0answers
38 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
0
votes
1answer
30 views

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational?

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational? That is, when is it a root of a quadratic polynomial with integer coefficients? I believe that the answer ...
0
votes
3answers
36 views

Complex Number Geometry

Problem: There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $2+2i$, $5+i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1c_2$. I've been struggling with ...
0
votes
3answers
69 views

The $n^{th}$ root of $1$

How do I prove that the $n^{th}$ root of $1$ forms the vertices of a unit $n$-gon in the complex plane? For example when I graph the $3$ solutions of $\sqrt[3]{1}$ they form the vertices of a unit ...
0
votes
2answers
66 views

Understanding proof for $\sum_{k=1}^{n-1}{\sin{\frac{2\pi k}{n}}} = 0$

(English is not my native language, so I apologize if I fail to use the right technical terms) I am stuck in proving the following. I'll explain how far I got and maybe someone can help me out by ...
1
vote
1answer
55 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
1
vote
3answers
45 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
4
votes
2answers
176 views

Discriminant of $x^n-1$

Question is to find discriminant of polynomial $x^n-1$ I consider $f(x)=x^n-1=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)$ Now, ...
2
votes
0answers
67 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
vote
1answer
48 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. ...
5
votes
0answers
51 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
vote
0answers
33 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
votes
2answers
36 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
votes
1answer
32 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 ...
5
votes
0answers
46 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
vote
1answer
58 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
vote
2answers
53 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
votes
1answer
47 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
31 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
votes
3answers
54 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
273 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
votes
3answers
80 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
votes
1answer
59 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 ...
1
vote
0answers
46 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 ...
3
votes
0answers
70 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
153 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 ...
3
votes
3answers
144 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 ...