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Questions tagged [roots-of-unity]

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

47
votes
3answers
15k 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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
30
votes
1answer
427 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 ...
26
votes
7answers
14k 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-...
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 $X_N$ ...
18
votes
3answers
2k 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 ...
16
votes
7answers
5k 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 $...
11
votes
3answers
4k views

Quadratic subfield of cyclotomic field [duplicate]

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}(...
11
votes
1answer
300 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 ...
11
votes
1answer
987 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 ...
10
votes
5answers
2k views

Product of one minus the tenth roots of unity [closed]

If $1$, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots \alpha_9$ are the $10$th roots of unity, then what is the value of $$ (1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3) \cdots (1 - \alpha_9)? $$ I am ...
10
votes
1answer
499 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 ...
9
votes
4answers
3k 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 ...
9
votes
1answer
379 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. Given a prime $p=6m+1$. Define, $$F(x) = ...
8
votes
1answer
230 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
315 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 ...
8
votes
1answer
234 views

Find $P(x,y,z)=x^n+y^n+z^n-\prod\limits_{k=0}^{n-1}(x+\omega_n^ky+\omega_n^{-k}z)$, where $\omega_n$ denotes a primitive $n$th root of unity

Find $P(x,y,z)=x^n+y^n+z^n-\prod\limits_{k=0}^{n-1}(x+\omega_n^ky+\omega_n^{-k}z)$, where $\omega_n$ denotes a primitive $n$th root of unity. I have manually multiplied the terms of the product and ...
7
votes
1answer
995 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.
7
votes
2answers
168 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
331 views

Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

For each positive integer $n$, let's define the polynomial $$P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$$ Do the zeroes of $P_n$ lie inside, outside, or on the unit circle $|z|=1$? I tried ...
7
votes
1answer
577 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
3answers
152 views

Sum of $1+4\epsilon +9\epsilon^2 +16\epsilon^3+…+2018^2 \epsilon^{2017}$

I have this problem from a college exam: Let $\epsilon$ be a non-real root of unity of order 2018, find the sum$$S=1+4\epsilon +9\epsilon^2 +16\epsilon^3+...+2018^2 \epsilon^{2017}$$Here is my try. ...
7
votes
3answers
381 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 $...
7
votes
2answers
195 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{...
7
votes
1answer
293 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 ...
7
votes
2answers
288 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
874 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 ...
7
votes
1answer
119 views

Complex numbers involving roots of unity

Let $z\in \mathbb{C}$ and $n\in \mathbb{N}, n \geq 1$. Solve the following equation: $$z+z^2+\dots+z^n=n|z|^n$$ Obviously, $(z,n)=(0,n)$ and $(1,n)$ are solutions, $\forall n \geq 1$. Considering $z \...
7
votes
0answers
198 views

Sum of binomial coefficients in Gould tables

Consider the combinatorial identity by Gould, Table III, page 25, equation (6.13): $$\sum_{k=0}^{[\frac{n}{r}]}{n \choose rk}=\frac{2^n}{r}\sum_{j=1}^{r}\left(\cos{\frac{\pi j}{r}}\right)^n\cos{\frac{...
6
votes
2answers
667 views

$ \bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

Is $$ \bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$$my argument is that the argument of the elements of the first set are rational multiples of $\pi$ whereas the second set also ...
6
votes
4answers
720 views

Moving from $x^2+x+1=0$ to $x+1+\frac1x=0$ seems to introduce a solution, how is this possible?

I recently saw this: ($x\in\mathbb{C}$) $$x^2+x+1=0,x\neq0,\pm1$$ $$x+1+\frac{1}{x}=0$$ Plugging (2) into (1) we get $x^2-\frac{1}{x}=0$ so $x=1$ Why does this happen? I know that eq. (1)...
6
votes
1answer
127 views

How to exactly determine whether a sum of n-th roots of unity is zero

Define the set $R = \{e^{2\pi i k/n} | k=0,1,\ldots,n-1\}$ of $n$-th roots of unity. Let $S \subseteq R$ be a subset. How can I (algorithmically?) determine whether $\sum_{s\in S} s = 0$? I'm ...
6
votes
4answers
151 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
2answers
947 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 ...
6
votes
1answer
76 views

“Converse” to the theorem “sum of roots of unity equal 0”

It is well known that sum of roots of unity equal 0. However, if $\sum_j \exp(i \phi_j)=0$, can we say something about the relation between the $\exp(i \phi_j)$'s? For example, we can rotate one of ...
6
votes
1answer
461 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 $\zeta$ ...
6
votes
0answers
86 views

Rational Trig Solutions for $n\ge 3$

Are there solutions to $$\sin(x+y)\sin(x-y)=n\ \sin(x)\sin(y)$$ for $n\ge 3$ where $x$ and $y$ are rational multiples of $\pi$? (excluding the trivial solutions when both sides are $0$). Known ...
6
votes
0answers
148 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 ...
5
votes
3answers
521 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 ...
5
votes
3answers
278 views

Concurrent lines proof for a regular 18-gon

Let $X_1 X_2 \dotsb X_{18}$ be a regular 18-gon. Show that $X_1 X_{10}$, $X_2 X_{13}$, and $X_3 X_{15}$ are concurrent. What would be the best way to prove this? I am actually struggling ...
5
votes
1answer
6k views

Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$

Determine the splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$ Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
5
votes
3answers
1k views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
5
votes
4answers
204 views

Minimum polynomial of a root involving the 7th root of unity

Let ω be a primitive 7th root of unity in $\Bbb C$ and set $α := ω + ω ^6$ . Determine, with justification, the minimum polynomial of α over Q. Would one use logs in such a question , or how should ...
5
votes
3answers
3k 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 ...
5
votes
3answers
378 views

How to express a power series in closed form

I am trying to express the power series $x + x^4/4! + x^7/7! + \cdots$ in closed form; I have already worked out the power series $1 + x^3/3! + x^6/6! + \cdots$ to be $(e^x + e^{x(2\pi i/3)} + e^{x(2\...
5
votes
5answers
410 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
202 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 $\sqrt{n}$...
5
votes
3answers
458 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 ...
5
votes
1answer
1k views

Long integer multiplication using FFT in integer rings

I would like to perform long integer (~= polynomial) multiplication using the FFT or its direct analogue, but never leave integer rings. Please excuse in advance all my mistakes in formulation and ...
5
votes
3answers
162 views

if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then…

if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then $$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$ ...
5
votes
1answer
115 views

Simple proof of existence of number field extension with the same roots of unity?

I'm looking for a brief proof of the following statement: Let $K$ be a number field and $p$ a prime number. Then there exists a field extension $L\supset K$ of degree $p$ with $\mu(L)=\mu(K)$. My ...