# 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

636 questions
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### Another Roots of Unity Sum

I almost see a brute-force attack on this problem, but before messing with the details I wonder there is some theory here, or at least a nice way to group the terms so I can see the cancellation. Let ...
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### Direct product decomposition of the group of complex roots of unity

I'm studying $p$-adic numbers (Robert's "A course in $p$-adic analysis) and, at page 41, the author states that, for every prime $p$, the group $\mu$ of all complex roots of unity has a direct product ...
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### $a_i$ are the n-th roots of $1\in\mathbb{C}$, why does $(1-a_2)\cdot…\cdot(1-a_n)=n$?

For $1<i\leq n$, let $a_i$ be the n-th roots of $1\in\mathbb{C}$, why does $(1-a_2)\cdot...\cdot(1-a_n)=n$?
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### Question about roots of unity in the Fast Fourier Transform

I am learning about the Fast Fourier Transform, which converts a polynomial from its coefficient representation into its point-wise form using divide-and-conquer. The Fast Fourier Transform evaluates ...
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### Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}$

Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}.$ I'm trying to use Dirichlet's Unit Theorem to solve this problem. It states that if $K$ is a number ...
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### Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
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### Why are roots of unity evenly spaced?

Roots of unity are the solutions of the complex polynomial $t^{n}-1=0$ they have the following form $E_{n}=\{e^{\frac{2\pi ik}{n}}:k\in\mathbb{Z}\}=\{e^{\frac{2\pi ik}{n}}:k=1,...,n-1\}$. From the ...
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### (Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Here is the work I've done so far: $\sqrt[p]{2}$ is a real root of $f(x)$ Any $(\zeta \sqrt[p]{2})$ where $\zeta$ is a $p^{th}$ root of unity is also a root of $f(x)$ Since $p$ is prime, all its ...
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### Why is $\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$ independent of $\alpha$?

By accident I found (numerically) that the expression $$\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$$ only depends on $\beta$. This looks like it shouldn't be ...
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### $\bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

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 ...
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### $\zeta_n^k$ is primitive if and only if $(k,n) = 1$ [duplicate]

Show that for $k \in \mathbb{Z}$: Is $\zeta_n$ a primitive $n$-th root of unity, then $\zeta_n^k$ is primitive if and only if $(k,n) = 1$. I only need the backwards direction: $\zeta_n^k$ is ...
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### How to get common roots of unity of $z^{6}=1$ and $z^{21}=1$?

How to get common roots of unity of $z^{21}=1$ and $z^{6}=1$ for $z\in\mathbb C$? I know that $z^{n} =1$ has roots $z=e^{\frac{2\pi k }{n}i}$ where $k\in \{0,1,2,...,n-1\}$
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### Determine a Normal Basis for Galois Extension of $\mathbb{Q}$ with primitive pth root of unit (p prime)

Let p be a prime, $\xi_p \in \mathbb{C}$ a primitive p-th unit root and $K = \mathbb{Q}(\xi_p)$. Give a normal basis for $K/\mathbb{Q}$. I know, that a basis of $L/K$ (finite and galois) is ...
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### If $y^4 = 5$ and $z^6 = 15$, then $y \notin \mathbb{Q}(z)$

Let $y,z \in \mathbb{C}$ with $y^4 = 5$ and $z^6 = 15$. I want to show that $y \notin \mathbb{Q}(z)$. So we have $$y = w_1 \cdot \sqrt5 \;\;\;\;\;\;\; z = w_2 \cdot \sqrt{15}$$ with $w_1$ ...
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### What is the sum of the squares of the 10th roots of unity?

Obviously the sum of the roots of unity is 0, but is there a way to calculate this other than calculating them all individually and squaring them?
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### 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 ...
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### “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 ...
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### cardinality of Galois group in $\mathbb{Q}$($\zeta_n$) [duplicate]

Let $\mathbb{Q}$($\zeta_n$) be some cyclotomic field, where $\zeta_n$ is a n-th root of unity. I already managed to show that $\mathbb{Q}$($\zeta_n$) is an Galois extension, but now i struggle to ...
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### 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 $$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$ ...
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### Number of roots of unity on the unit circle

I have to find the number of roots of unity on the unit circle |z|=1 in the argand plane. I know that there are n roots of the the equation $z^n=1$ and all of them lie on the given circle. Does that ...
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### $\zeta$ is a p-th primitive untary root iff $-\zeta$ if a 2p-th primitive unitary root, with $p$ an odd prime

$\Rightarrow$ $(-\zeta)^{2p}=(-\zeta^p)^{2}=1$ and if $i \in[{1,2p-1}]$ there is $k \in[0,p-1]$ such that $i=2k+1$ if is odd and $2k$ if is even so $(-\zeta)^{i}=(-\zeta)^{2k}=(\zeta^k)^{2}\neq 1$ ...
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### Prove that sum of n-th degree roots of complex number is 0

I'm trying to prove, that sum of all complex roots of n-th degree of a complex number $z$ is equal to 0. I know how to prove it for $z = 1$ (roots of unity), however i have to prove it for any complex ...
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### 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 ...
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### About roots of unity in an arbitrary CM field K with an abelian extension L

I have an arbitrary CM field $K$ and an abelian extension $L$ of $K$ with dimension $[L:K]=p$, with $p$ prim. Suppose there exist a non trivial primitiv root of unity $\xi$ in K. Must $\xi$ be a $p$-...
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### Rings with a finite set of units

Consider a ring $\mathcal{R}$ with a finite set $\mathcal{R} ^\times$ of units, i.e. divisors of $1$, for example $\mathbb{Z}^\times = \{\pm 1\}$ $\mathbb{Z}[i]^\times = \{\pm 1,\pm i\}$ (Gaussian ...
### Quadratic Gauss sum using $|Z|^2=Z \bar Z$
Given $Z=\sum_{k=1}^{n-1}\omega^{k^2}$ I'm asked to find $|Z|^2$, here's what I thought of: |Z|^2=Z \bar Z=\left( \sum_{k=1}^{n-1}\omega^{k^2} \right) \left( \sum_{k=1}^{n-1}\frac{1}{\omega^{k^2}} \...
I just cannot seem to wrap my head around this problem and would really appreciate some guidance. So I know that a primitive $n^{th}$ root of unity is a complex number $z$ such that $z^n = 1$ but \$z^m ...