Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?

Let $p,q$ be (odd) primes, and let $m,n,s,t$ be positive integers. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. M. A. Nyblom showed that, if $s = 4m - 3$...
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Some questions on cyclotomic polynomials

I observed that different books mention properties of cyclotomic polynomials in terms of lemmas (not obvious proof) and some books mention same thing as obvious thing. Here are two of them. Can one ...
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Order and Structure of Galois group

I'm struggling with a question I've got as I can't seem to wrap my head around Galois theory much. i)Prove that $\zeta^{2}$ is a primitive 8th root of unity and $\zeta^{4}$ is a primitive 4th root of ...
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What can be said about the primality of Zsigmondy Numbers?

Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ by the $n$-th Zsigmondy number to base $(a,b)$, where $\Phi_n(a,b)$ is the $n$-th homogeneous cyclotomic polynomial. Zsigmondy proved ...
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How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreductible in $\mathbb{F}_{2}$

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreducible over $\mathbb{F}_{2}$. The main 2 "weapons" I have at my disposal is the Eisenstein criteria and reduction criteria but neither seem to ...
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factoring a cyclotomic polynomial modulo a prime

I am having trouble with this: it appears that $x^{12} - x^6 + 1$ has a root $\pmod p$ only when $p \equiv 1 \pmod{36},$ in which case it completely factors as twelve linear terms. It is easy to ...
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Is $g(x,y)= \frac{f(x^{2y+1},y)}{f(x,y)}$ always an integer?

This question is similar to this other question: Let $$f(x,y):= \frac{x^y -1}{x+(-1)^y}$$ and $$g(x,y):= \frac{f(x^{2y+1},y)}{f(x,y)}.$$ Let $y\ge1$ be an integer. Show that $g(x,y)$ is a ...
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Cyclotomic polynomial with nonnegative coefficients

I know that the cyclotomic polynomial $\Phi_m = \prod (x-\zeta_m^c)$ for all $c$ coprime to $m$, with $\zeta_m$ a primitive $m^{th}$ root of unity is the unique minimal polynomial with integral ...
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Why is $\mathbb{Q}[\zeta_{p}] = \{a_{0}, a_{1}\zeta, \dots, a_{p-2}\zeta^{p-2}, a_{i} \in \mathbb{Q}\}$, where $p \geq 2$ is a prime?

Let $p$ be a prime number $\geq 2$ and $\zeta := \zeta_{p}$ a complex number where $\zeta \neq 1$ and $\zeta^p=1$, i.e., is a $p$th root of unity. This is a question from a brazillian book, Introdução ...
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Are there more perfect powers of this form?

Let $a,b$ be integers , $a\ge 3$ , $b\ge 1$ If $f_a$ is the $a$ th cyclotomic polynomial, are there any perfect powers of the form $f_a(b)$ apart from $121$ and $343$ ? In the range 3\le a\le 1\ ...
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What is the minimal polynomial of an $8$-th primitive root of unity over $\mathbb{Q}_3$?

Let $K = \mathbb{Q}_3$ and $\zeta_8$ a primitive $8$-th root of unity. Question: What is $\min_K(\zeta_8)$? I know that the extension $K(\zeta_8)/K$ is unramified of degree $2$, so the degree of the ...
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Does the Bunyakovsky conjecture apply for every cyclotomic polynomial?

The Bunyakovsky conjecture states that for every irreducible polynomial $\ f\in \mathbb Z[x]\$ with positive leading coefficient, there are infinite many primes of the form $\ f(m)\$ , where $\ m\$ ...
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Showing that the irreducible factors of the cyclotomic polynomial in $\mathbb{Q}[x]$ have the same degree.

For each $n\in \mathbb{N},\, \,$ $\phi_{n}(x):=\displaystyle \prod_{\substack{1\leq k\leq n\\ (k,n)=1}}(x-e^{\frac{2\pi i}{n}k})$ Show that the irreducible factors of $\phi_n$ in $\mathbb{Q}[x]$ have ...
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For $p$ prime and for $1 ≤ e ∈ Z$ the prime-power $p^e$-th cyclotomic polynomial $Φ_{p^e} (x)$ is irreducible in $Q[x]$

This is part of 'Abstract Algebra, Paul Garrett, 243-244p' Recall that $Φ_{p^e}(x) = Φ_p(x^{p^{e-1}}) = \frac{x^{p^{e}}-1}{x^{p^{e-1}}-1}$ First, we check that $p$ divides all but the highest-...
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Minimal polynomial of $e^{\frac{2 \pi i}{p^2}}$ (p is prime number)

$X^{p^2}-1 = (X-1)(X^{p-1}+X^{p-2}+..+X+1)(X^{p(p-1)}+X^{p(p-2)}+..X^{p}+1)$ Then, $e^{\frac{2 \pi i}{p^2}}$ is a root of $f(X)=X^{p(p-1)}+X^{p(p-2)}+..X^{p}+1$ Hence, $g(X)=\frac{X^{p^2}-1}{X-1}$ for ...
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What is the $n$th Cyclotomic Polynomial for a given $n$
Are there any formula to compute the $n$th cyclotomic polynomial for a given $n$? I know how to compute it for small $n$, but for large $n$, I need a general formula..