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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2
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0answers
31 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 ...
3
votes
3answers
295 views

A Polygon is inscribed in a circle $\Gamma$

A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove ...
1
vote
3answers
164 views

Subtracting roots of unity. Specifically $\omega^3 - \omega^2$

This is question that came up in one of the past papers I have been doing for my exams. Its says that if $\omega=\cos(\pi/5)+i\sin(\pi/5)$. What is $\omega^3-\omega^2$. I can find $\omega^3$ and $\...
1
vote
1answer
46 views

What is meant by a number to be a root of unity?

I am proving first case of Fermat's last theorem for regular primes by following Marcus' book "Number Fields". I have to prove following statement: If $\varepsilon$ is a unit in $\mathbb{Z}[\omega]...
4
votes
2answers
74 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 ...
2
votes
0answers
51 views

lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set $S = \{ |s(\vec x)| : \vec x \...
6
votes
2answers
93 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?
0
votes
0answers
29 views

Trace of roots of unity has valuation more than 1

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
0
votes
1answer
24 views

Finding primitive root of unity using Newton iteration

following problem I am supposed to solve on paper. Use Newton-Iteration to find a primitive 16th root of unity over $\mathbb{Z}/17^{16}\mathbb{Z}$ with $\omega = 6 \pmod{17}$ I have some kind ...
0
votes
2answers
48 views

A sum of powers of primitive roots of unity

For the primitive roots of unity $\omega_n = e^{i2\pi/n}$ I want to prove that $$\sum_{k=0}^{n-1} \omega_n^{lk} = 0$$ if $n$ doesn't divide $l$. I have already proven the well-known result $$\sum_{k=...
0
votes
1answer
17 views

Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
1
vote
2answers
22 views

Number of roots of a complex exponent

There are $p$ solutions to $\sqrt[\frac{p}q]1$, if $\frac{p}q$ is a fraction in lowest terms. I have found on this website that an irrational exponent has infinite roots. But what about $\sqrt[a+bi]1$...
2
votes
1answer
35 views

Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
3
votes
1answer
62 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
2
votes
1answer
37 views

Sum of cosines with a multiplicative factor in the angle and different interval

I have found the following formula for the sum of cosines in both here and here. \begin{align} \sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0 \end{align} I would like to know what the sum ...
1
vote
2answers
46 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
0
votes
1answer
18 views

Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$.

I'm starting to see complex numbers in algebra. I've missed a few classes and I have exercises similar to this one: Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$. ...
3
votes
4answers
120 views

minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...
3
votes
2answers
128 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
2answers
46 views

Problem based on sum of reciprocal of $n^{th}$ roots of unity

Let $1,x_{1},x_{2},x_{3},\ldots,x_{n-1}$ be the $\bf{n^{th}}$ roots of unity. Find: $$\frac{1}{1-x_{1}}+\frac{1}{1-x_{2}}+......+\frac{1}{1-x_{n-1}}$$ $\bf{My\; Try::}$ Given $x=(1)^{\frac{1}{n}}\...
1
vote
6answers
402 views

Sum of nth roots of unity

Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$ Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}...
2
votes
1answer
65 views

Galois correspondence for the field extension $\mathbb{Q}(\omega_7)$

Let $E = \mathbb{Q}(\omega_7)$, where $\omega_7$ is the 7th root of unity. We know that $$\mathbb{Q}(\omega_7) \cong \mathbb{Z}_7^{\times},$$ where $\mathbb{Z}_7$ is the multiplicative group of units, ...
1
vote
2answers
42 views

Polynomial roots. Finite Extensions.

If $a,b,c$ belong to $\mathbb Q$ and $ω^3=1$ is a third root of unity, then prove that if $$(a+b\sqrt[3]{2}+c\sqrt[3]{4})^3=1+2\sqrt[3]{2}-\sqrt[3]{4}$$ holds we also have $$ 1+2ω\sqrt[3]{2}-\bar ω\...
0
votes
3answers
112 views

If $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + … + \omega^{n-1} = 0 $? [duplicate]

Let $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + ... + \omega^{n-1} = 0 $? I saw this on a algebra PPT slice. However the teacher did not explain why this equation is correct, can ...
1
vote
1answer
34 views

root of unity in quotient field

Let R be an commutative ring, in which $n \ge 2$ is invertible. Show that the modulo class of $y$ in the quotient ring $ R'=R[y]/\langle y^n-1\rangle $ is not an primitiv $n$th-root of unity in R'. ...
0
votes
1answer
23 views

roots of modular forms in the complex field

For $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z})$ the modular discriminant $$\Delta(z)=(2\pi)^{12}\eta(z)^{24}\qquad(1)$$ holds $$\Delta\left(\dfrac{az+b}{cz+d}\right)=(cz+d)^{12}\...
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 $r$ ...
2
votes
2answers
87 views

If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$

then $(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$ equals to ? I think here we need the info of whether $n$ is even or odd else how will we say whether by vieta's formula what is the value of $1+(-1)^n$...
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
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 ...
2
votes
1answer
55 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
0
votes
0answers
12 views

Complex Number Powers of Coprime Rational Powers

I'm trying to figure out $z^{p/q}$ where $p,q$ are coprime. Suppose I want to find $z^{2/7}$ where $z=128$. I can rewrite $z=128e^{0}$ Now I know that the $z^{1/7}$ roots are $2e^{k2\pi i/7}$ for $...
4
votes
0answers
85 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
4
votes
1answer
71 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
0
votes
1answer
20 views

Proof that $t^m=t^{j}$ if $t$ is an $r^{th}$ root of unity such that $r \mid k$.

I need help with the following proof. Let $j$ = $0,1,\ldots, k-1$. Also, let $t$ be an $r$th root of unity other than $t=1$ such that $r \mid k$. We know $m=j\pmod k$. Furthermore, $m$, $j$ and $k$ ...
0
votes
1answer
52 views

Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ [duplicate]

Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ We must show that it is normal and separable. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it ...
0
votes
1answer
61 views

How many fields are there between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_{15})$?

$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that: $\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$ Is there a ...
2
votes
2answers
74 views

Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

Suppose $ \zeta$ is a primitive $ 11$-th root of unity and $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $ Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ Could someone please give me a hint ...
1
vote
3answers
68 views

Sum of n-th roots of unity [duplicate]

I'm being asked to prove that $$1 + \omega + \omega^2 + ... + \omega^{n-1} = 0$$ where $\omega \ne 1$ is an n-th root of unity, and I don't know where to start I feel like there's something terribly ...
1
vote
3answers
108 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
2
votes
1answer
40 views

Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
3
votes
0answers
32 views

Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
2
votes
2answers
147 views

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. ...
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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
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-...
1
vote
0answers
30 views

Roots of unity in fixed field of decomposition group.

$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...
0
votes
0answers
19 views

Norm of roots of unity in a subextension.

Suppose $L/k$ is a Galois extension of number fields, $G$ is the Galois group, and for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$. Suppose $m$ is the ...
0
votes
1answer
11 views

$f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$ - Dense on the unit circle

Personal question : Let $z_0$ be a $2^k$th root of unity. We obtain for the function $f(z)=\sum_{n \geq 0} z^{2^n}$ (radius of convergence $R=1$) that $f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$. Why ...
0
votes
1answer
26 views

The norm and the angle of the complex number $\sqrt[3]{-1}\sqrt[6]{7}$?

I am learning the roots of unity here. I want to express arbitrary complex number in radical form in terms of the norm and the angle to use the formula $e^{i\phi}=\cos(\phi)+i\sin(\phi)$. Consider $x^...
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 ...