Tagged Questions

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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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 ...
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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 ...
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$\mathbb Z$ 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 ...
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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 \...
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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?
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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 ...
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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 ...
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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 $\sqrt[a+bi]1$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $w \in G_7$. ...
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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 ...
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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 $\mathbb C$)...
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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 ...
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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 ...
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$f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$ - 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 ...
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^...