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
\begin{equation}
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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How does $x^n-1$ Factor in an Arbitrary Field?
It's well-known that over $\Bbb Q$ the factorization into irreducible cyclotomic polynomials$$x^n-1=\prod_{d|n}\Phi_d(x)$$and over $\Bbb F_{p^m}$ it's well-known as well from basic Galois theory that $...
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Fifth cyclotomic polynomial over a finite field
Consider the polynomial $g(x)=x^4+x^3+x^2+x+1 \in \mathbb{F}_3[x]$. It's possible to show that $g$ is irreducible in $\mathbb{F}_3$. If we let $\alpha$ be a root of $g$, then $\alpha^4+\alpha^3+\alpha^...
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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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