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Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

If $n$ is a positive integer, the $n$th cyclotomic polynomial is defined to be the unique irreducible polynomial with integer coefficients which is a divisor of $x^n - 1$, but not of $x^k - 1$ for any $0 < k < n$.

Alternatively, the $n$th cyclotomic polynomial can be written as

$$\Phi_{n}(x) = \prod_{\stackrel{1\le k\le n}{\gcd(k,n)=1}} (x - e^{2i\pi \frac{k}{n}})$$

Source: Cyclotomic polynomial.

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showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, without proof, that $n$th cyclotomic polynomial…
NNNN
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How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex numbers all have the form…
Bruno Joyal
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Elementary proof of Zsigmondy's theorem

I've been writing a not-so-short introduction to elementary number theory, supplying proofs for all theorems. When coming across Zsigmondy's theorem, it seemed difficult to find a proof available on the internet. I've been searching through MSE's…
Bart Michels
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Series of Cyclotomic polynomials

How can I show that if $\Phi$ is a Cyclotomic polynomial, $$\Phi_n(x)=\prod_{\substack{1\leq k\leq n\\(n,k)=1}}(x-\zeta_n^k)$$ With $\frac{d}{dx}\Phi_n(x)=\Phi'_n(x)$ Then,…
Ethan Splaver
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Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The irreducibility of its other factor,…
Lucian
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What comes after $\cos\left(\tfrac{2\pi}{7}\right)^{1/3}+\cos\left(\tfrac{4\pi}{7}\right)^{1/3}+\cos\left(\tfrac{6\pi}{7}\right)^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ $$\big(\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}\big)^3 =…
Tito Piezas III
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Is the "cyclotomic diagonalization" always squarefree?

For every integer $n\ge 2$ , define $$f(n):=\Phi_n(n)$$ where $\Phi(n)$ is the $n$ th cyclotomic polynomial. This can be considered as the "cyclotomic diagonalization" Prove or disprove the conjecture that $f(n)$ is squarefree for every integer…
Peter
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Coefficient of $n$th cyclotomic polynomial equals $-\mu(n)$

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that $$\varphi(\Phi_n)=-\mu(n)$$ where $\mu(n)$ is the Möbius…
user140049
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Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that $\bigl(X(X-a)\bigr)^{2^n} +1$ is irreducible over…
user84673
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What is the number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because $a-b$ can't be factored further. $f(2)=2$,…
wythagoras
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Finite-order elements of $\text{GL}_4(\mathbb{Q})$

I'm currently studying for my qualifying exams in algebra, and I have not been able to solve the following problem: Determine all possible positive integers $n$ such that there exists an element in $\text{GL}_4(\mathbb{Q})$ of order $n$. I've been…
Debbie
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Derivatives of the nth cyclotomic polynomial

Are there any useful properties of the $k$th derivative of the $n$th cyclotomic polynomial? In particular, what would the value of this be at $1$ and $0$, or any properties of the value of the $k$th derivative at $0$ or $1$?
Mayank Pandey
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When is $(x^n-1)/(x-1)$ a prime number?

Let $x > 1$ and let $n$ be a prime. I'm wondering if a characterization of this is known. That is, what are sufficient and necessary conditions for $$ \dfrac{x^n-1}{x-1} = 1 + x + x^2 + \cdots + x^{n-1} $$ to be a prime number? What are these…
user152634
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Is there a nice general formula for $\int \frac{dx}{x^n-1}$ and/or $\int \frac{dx}{\Phi_n(x)}$?

Question is in the title, where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. Motivation: I'm just teaching my calculus students basic integration of rational functions with $\log$ and $\arctan$, so I wondered about this. Observation and more…
Torsten Schoeneberg
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Discriminant of cyclotomic polynomial $\Phi_p(x)$

I don't know tricks for computing the discriminant of a polynomial, only the definition and using the resultant, but it's very complicated to do only with that tools. I need some help please. I have to prove that the discriminant of $\Phi_p$ is…
Daniel
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