# Tagged Questions

For questions related to cyclotomic polynomials and their properties.

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### Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
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### how can I find the smallest integer n such that a polynomial divides x^n-1

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example ...
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### Computation of a certain integral involving cyclotomics

How would one compute $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz$ in terms of k and n. If this is not possible, how would someone find a good approximation for this. ...
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### Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
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### Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
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### Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)

With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where ...
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### Points on a unit circle

Let $P_1, P_2,..., P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2,3,...,2013\}$ is the product of all pairwise distances: $$\prod_{1\le i\lt j\le n} P_{i}P_{j}$$ a ...
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### Factorization of seventh cyclotomic polynomial

The fifth cyclotomic polynomial $\Phi_5(z)$ factors as $$\Phi_5(z)=(z^2+\varphi z+1)(z^2+(1-\varphi)z+1)$$ where $\varphi$ and $1-\varphi$ are the solutions to $x^2-x-1$. Of course, $\varphi$ is ...
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### Isomorphism betwixt a Galois Group and the set of $n$th roots of unity

Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...
### The cyclotomic polynomial $g_m$ is irreducible over $\mathbb{Z}_p$ if and only if $U(\mathbb{Z}_m)$ is a cyclic group generated by $\bar{p}$.
Let $p$ be a prime which does not divide $m$ and $\alpha$ be a primitive $m^{th}$ root of unity over $\mathbb{Z}_p$. Let $k$ be the order of $\bar{p}=p (\mod m)$ in the group of units in ...
Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ ...