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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29
votes
1answer
366 views

A spiralling sequence based on integer divisors. Has anyone noticed this before?

Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms. I have mapped integers to points on a ...
27
votes
3answers
6k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
26
votes
2answers
1k views

Drunkard's walk on the $n^{th}$ roots of unity.

Fix an integer $n\geq 2$. Suppose we start at the origin in the complex plane, and on each step we choose an $n^{th}$ root of unity at random, and go $1$ unit distance in that direction. Let $X_N$ ...
20
votes
6answers
8k views

Intuitive understanding of why the sum of nth roots of unity is $0$

Wikipedia says that it is intuitively obvious that the sum of $n$th roots of unity is $0$. To me it seems more obvious when considering the fact that $\displaystyle 1+x+x^2+...+x^{n-1}=\frac{x^n-1}{x-...
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 $...
17
votes
3answers
1k views

Is $\sqrt 7$ the sum of roots of unity?

Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals. Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ? How to ...
10
votes
1answer
316 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
10
votes
1answer
213 views

Roots of unity in non-Abelian groups: when do they form subgroups?

I haven't studied group theory in earnest beyond first courses, so my notation may be nonstandard and my question may be a 'standard fact', so bear with me: Consider a group $G$, and for each natural ...
10
votes
1answer
519 views

Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
9
votes
1answer
266 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-mx+N = \Big(x-2\sum_{k=...
8
votes
1answer
161 views

Inverse Limits: Isomorphism between Gal$(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q})$ and $\varprojlim (\mathbb{Z}/n\mathbb{Z})^\times$

I'm trying to prove that $\operatorname{Gal}(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q}) \cong \widehat{\mathbb{Z}}^\times = (\varprojlim (\mathbb{Z}/n\mathbb{Z}))^\times$, where $\varprojlim$ ...
8
votes
1answer
278 views

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...
7
votes
4answers
1k views

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
7
votes
2answers
148 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 ...
7
votes
1answer
197 views

Determining if a complex number is a root of unity

How would you determine if $a+ib$ is an nth root of unity? Obviously, the modulus of $a+ib$ must be $1$. But you would also need to determine if $a+ib$ is located at a vertex of a regular ...
7
votes
2answers
275 views

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let $\...
7
votes
1answer
692 views

Zero sum of roots of unity decomposition

It's known that sum of all $n$'th roots of some $z \in \mathbb C$ with $|z| = 1$ is zero (if $n \geqslant 2$). Is it true that any zero sum of roots of unity can be decomposed in this way? That is if ...
6
votes
3answers
2k views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be $\mathbb{Q}(...
6
votes
3answers
174 views

Showing that $\mathbb{Q}(\zeta_p, \sqrt[p]{\ell}) = \mathbb{Q}(\zeta_p + \sqrt[p]{\ell})$ for $p,\ell$ primes.

We consider the polynomial $x^p - \ell$, where $p,\ell$ are both prime numbers. Let $\zeta_p$ be a $p$-th root of unity. We wish to show that $L = \mathbb{Q}(\zeta_p, \sqrt[p]{\ell})$ is the same as $...
6
votes
2answers
92 views

Factorization of $x^n + y^n$, what sort of coefficients show up?

We know that$$a^2 + b^2 = (a + bi)(a - bi).$$ What are the complete factorizations of $a^3 + b^3$, $a^4 + b^4$, $\ldots$ , $a^k + b^k$, etc.? What sort of coefficients show up?
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 ...
6
votes
1answer
186 views

What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p $ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
6
votes
1answer
290 views

Relation that holds for the Legendre symbol of an integer but not for the Jacobi symbol?

Let $p$ be a prime number and $\big(\frac{a}{p} \big)$ the Legendre symbol. Then we have the equality $$\sum_{a=1}^{p-1} \big(\frac{a}{p} \big) \zeta^a =\sum_{t=0}^{p-1} \zeta^{t^2},$$ where $\zeta$ ...
5
votes
1answer
445 views

Why can't we just say 1 instead of “unity”?

I know this is a soft question of sorts but I am curious why we can't just say "1" instead of "unity," e.g. a root of unity.
5
votes
3answers
114 views

Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$

Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$ Here is what I tried: First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$. Let $r$ ...
5
votes
5answers
367 views

Prove that $(x^2-x^3)(x^4-x) = \sqrt{5}$, where $x= \cos(2\pi/5)+i\sin(2\pi/5)$

Prove $(x^2-x^3)(x^4-x) = \sqrt{5}$ if $x= \cos(2\pi/5)+i\sin(2\pi/5)$. I have tried it by substituting $x = \exp(2i\pi/5)$ but it is getting complicated.
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{...
5
votes
1answer
689 views

A finite field extension $K\supset\mathbb Q$ contains finitely many roots of unity

I must show that any finite field extension $K\supset \mathbb Q$ can only contain finitely many roots of unity. I reasoned in the following manner: Let $n<\infty$ be the degree of $K$ over $\...
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 ...
5
votes
2answers
301 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? ...
5
votes
2answers
603 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
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}\\ \...
5
votes
1answer
63 views

Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
5
votes
0answers
67 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 ...
5
votes
0answers
64 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 $\mathbb{...
4
votes
3answers
373 views

Question on primitive roots of unity

Let $p$ be an odd prime and $\omega$ be a primitive $p$th root of unity. The question is to prove that: $$(1-\omega)(1-\omega^2) \cdots (1-\omega^{p-1})=p$$ What I have done so far is: I can see ...
4
votes
2answers
177 views

Question on roots of unity

This may seem absurd but what is wrong with the next reasoning about $n$th roots of unit?. For $k,l\in\mathbb Z$ such that $0 \leq k < l \leq n-1$: $$ e^{2\pi i k/n} = (e^{\pi i})^{2 k /n} \overset{...
4
votes
2answers
73 views

$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
4
votes
2answers
109 views

is $\sqrt p$ in $\mathbb Q(\zeta_{4p})$?

i think for every prime $p$ we have $\sqrt p \in \mathbb Q(\zeta_{4p})$ when $\zeta_{4p}$ is a primitive 4p-th root of unity.but i have no idea to prove it. is it true? can any one help me with a ...
4
votes
2answers
201 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: $$\sqrt[3]{-1}=...
4
votes
1answer
138 views

A property of power series and the q-th roots of unity

I'm trying to understand why if $ \displaystyle \sum_{n=0}^{\infty} a_{n}x^{n} = f(x) $, then $$ \sum_{n=0}^{\infty} a_{p+nq} x^{p+nq} = \frac{1}{q} \sum_{j=0}^{q-1} \omega^{-jp} f(\omega^{j} x)$$ ...
4
votes
3answers
979 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 ...
4
votes
2answers
1k views

If z is one of the fifth roots of unity, not 1…

If z is one of the fifth roots of unity, not 1, show that: $1+z+z^2+z^3+z^4=0$ Which wasn't too bad, but the next part is killing me: show that: $z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$ Can ...
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 ...
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:...
4
votes
3answers
67 views

Alternating sum of roots of unity $\sum_{k=0}^{n-1}(-1)^k\omega^k$

Consider the roots of unity of $z^n = 1$, say $1, \omega, \ldots, \omega^{n-1}$ where $\omega = e^{i\frac{2\pi}n}$. It is a well known result that $\sum_{k=0}^{n-1}\omega^k = 0$, but what if we want ...
4
votes
2answers
98 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 ...
4
votes
2answers
64 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 $\bar{\mathbb{...
4
votes
4answers
147 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 ...
4
votes
1answer
108 views

Alternating Series Using Other Roots of Unity

$\sum (-1)^n b_n$ is representative of an alternating series. We look at whether $\sum b_n$ converges and if $b_{n+1}<b_n$ $\forall n\in \mathbb{Z}$. What if our alternating series is of the form $...