# Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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### Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

Let $c_k(n)$ denote Ramanujan's sum, and $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Prove that $$\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$$ My attempt was to ...
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### definition of cyclotomic polynomials

The $n$th cyclotomic polynomial can be expressed via the Mobius function as follows: $$\Phi_n(x) = \prod_{\substack{1\le d\le n\\d\mid n}}(x^d - 1)^{\mu(\frac{n}{d})}$$ In every reference I have ...
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### About the solutions of $\dfrac{x^p - y^p}{x - y} = a^2+pb^2$

Using theorem $IV$ from this article, is possible to prove that when $p$ is a prime $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the equation $\dfrac{x^p - y^p}{x - y} = a^2+pb^2$ ever ...
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### Irreducibility of the $p^k$-th cyclotomic polynomial

I want to prove that the cyclotomic polynomial $\Phi_{p^k}$ is irreducible using Eisenstein (I know that every cyclotomic polynomial is irreducible, I am just trying this approach). I am exposing what ...
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### In polynomial quotient rings over $\mathbb{F}_2$, how to use Chinese Remainder Theorem to solve equations?

Suppose I work over the polynomial quotient ring $\mathbb{F}_2[x,y]/\langle x^m+1,y^n+1\rangle$, and I want to solve for polynomials $s[x,y]$ that simultaneously solve the equations a[...
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### Find a monic irreducible polynomial equivelent to $(x-x_1)(x-x_2)\Phi_m$

Find a monic irreducible polynomial $f(x) = (x - x_1) ... (x - x_n)$， $|x_1| > 1$ and $x_1$ is real, |x_2| < 1 and $x_2$ is real, $|x_j| = 1$ for all $j > 2$. And First, prove $n > 3$ ...
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### Cyclotomic polynomials reducible/irreducible over $F_p$.

I'm studying for my exam (tomorrow!) and I came across the following problem that I'm not sure how to approach. Give an example if possible, and briefly explain why your example works. If no such ...
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### In formula $x_k=\cos(2k \pi/n) +i \sin(2k \pi/n)$ why does $k$ goes from $0$ to $n-1$?

Formula is for finding roots of unity. I know to prove that it will work for any k,but can't see in the formula why k needs to be from $0...n-1$. Obviously equation $x^n -1=0$ has $n$ roots,but why ...
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### The cyclotomic polynomials satisfy $\psi_{pn}(X)= \psi_n(X^p)$ if $p|n$?

This is the first exercise in section 6.5 of Robert Ash's abstract algebra, I want to understand the given solution: Noting that $\psi_n(X^p)= \prod_{w_i} X^p- w_i$, the roots of $X^p −ω_i$ are the $p$...
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### Is the angle on on Cartesian coordinate system between dots of all complex roots of polynomial with real coefficients the same?

So far I realized that any polynomial with complex roots has the even number of complex roots.Because for every $(x-(a-b*i))$ there is $(x-(a+b*i))$ in order for coefficiants to be real.That ...
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### Showing the polynomial has integer coefficients

Show that $\Phi_n(X)$ has integer coefficients. The proofs here states that $$\Phi_n(X)=\frac{X^n-1}{\prod_{d|n,d\ne n}\Phi_d(X)}.$$ And by long division, they get $\Phi_n(X)\in \Bbb{Q}[X]$. However, ...
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### Interpolating polynomial for characteristic function of primitive Nth roots of unity among all Nth roots

In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose ...
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### Values of $\Phi_n(-1)$

Let $$\Phi_n(x) = \prod_{0<k\leq n, \gcd(k,n)=1}(x-e^{\frac{2\pi i k}{n}})$$ be the $n$-th cyclotomic polynomial. By observation it seems that cyclotomic polynomials when evaluated at $x=-1$ give ...
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### Product of all $n$th-Cyclotomic polynomials less than $m$

Has the following function been studied? I'm looking for a reference, or an answer displaying some identities and properties $$f_m(x) = \prod_{n=1}^{m}\Phi_n(x)$$ It looks like a factorial analog of ...
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### Discriminant of numbers

In a previous post on a $q$-analog of number theory I present the fact that any $q$-number can be written as a unique product of cyclotomic polynomials, similar to the fundamental theorem of ...
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### Cyclotomic polynomial as minimal polynomial

I'm in the process of learning Galois theory and got stuck on Wikipedia's alternative definition of the $n$th cyclotomic polynomial as the "minimal polynomial over the field of the rational ...
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### Polynomials of the form $X^n-c$

Is there any specific name for the polynomials of the form $X^n-c$ for some $n$ and $c$ is a complex number? They seem to be closely related to Cyclotomic polynomials.
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### Show that $Tr(A)$ is in the spectrum of $A+I$

If we have $p$ an odd prime number and the matrix $A$ $(p×p)$ with rational entries and we know that $\det(A^{p}+I)=0$, $\det(A+I)≠0$. How we can show $\operatorname{Tr}(A)$ is in the spectrum of $A+I$...
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### Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$
In my notation, $\mathbb{Z}_n$ = the integers modulo $n$. Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$. (The additive group). I thought about using cyclotomic ...