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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.

491 questions
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On even cyclotomic polynomials

Let $\Phi_n$ be the nth cyclotomic polynomial. I would like to show that if $4$ divides $n$, then $\Phi_n$ is even. Any idea ?
Gaussian
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Equality between two products

Good evening. I'm currently working on cyclotomic polynomials $\Phi_n$, for $n \in \mathbb{N}^*$. I've proved by Möbius inversion formula that : $$\forall x \in \mathbb{C}, \hspace{1mm} \Phi_n(x) = \prod_{d|n} (x^d-1)^{\mu(n/d)}$$ I would like to…
LexLarn
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Product of all primitive nth roots of unity is 1?

Is the product of all primitive nth roots of unity equal to 1? Equivalently, is $\Phi_n(0) = 1$ for $n>2$? More exactly, I'm trying to prove that $$x^{\phi(n)} \Phi_n(x^{-1}) = \Phi_n(x).$$ Since $$\Phi_n(x) = \prod_{\substack{k = 1 \\ (k,n) =…
Cois Monis
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How can we know the root of 8-th cylotomic polynomial $\mod 3$?

Suppose we have 8-th cyclotomic polynomial $\Phi_8(x)=x^4+1$ It is standard fact that any cyclotomic polynomial is irreducible over rational field. When the coefficient is over $\mathbb{Z}_3$ i.e., $\mod 3$, it can be factorized. How can we…
mallea
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Why are the coefficients of the cyclotomic polynomial symmetric?

Why are the coefficients of the cyclotomic polynomial symmetric ? $\Phi_n(x):=\frac{x^n-1}{\prod\limits_{d |n, d
user257
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Prove one of the cyclotomic polynomial identities

Let $\Phi_n(x)$ be the nth cyclotomic polynomial over $\mathbb Q$. $\Phi_n(x)=\frac{x^n-1}{\Pi_{d|n,d1, and $\Phi_1(x)=x-1$ Let $n=p_1^{r_1}...p_s^{r_s}$ with $p_i$ distinct numbers and $r_i>0$. Show that…
aregak
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cyclotomic polynomial $\Phi_{2n}(x)$

Question is: "Show that $\Phi_{2n}(x)=\Phi_n(-x)$ for all odd numbers $n>1$" I try to prove this as follow, $\prod_{d \mid 2n} \Phi_d(x) =x^{2n}-1 = (x^n-1)(x^n+1) = -(x^n-1)((-x)^n-1) = -\prod_{d \mid n} \Phi_d(x)\Phi_d(-x)$ Then I write, $…
corcia candy
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A question about the degree of sin(2π/n) over the rationals

In cyclotomic theory, $\cos (2\pi /n)$) is shown to have degree $\varphi (n)/2$ over the rationals $\mathbb{Q}$, while $\sin (2\pi /n)$has degree $\varphi (n)$ (as long as n is not divisible by 4).$Cos(2\pi /n)=(\zeta +{{\zeta }^{-1}})/2$ where…
sunshineghh
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Prove that $Φ_{nm}(x) = Φ_n(x^m)$ if every prime divisor of m is also divisor of n

Let $m$ and $n$ be natural numbers that every prime divisor of $m$ is also a divisor of $n$. We can define $Φ_{ab}(x)$ for every prime $a>0$ like this: $$Φ_{ab}(x) = \begin{cases} Φ_b(x^a), & \text{if a|b} \\ \frac{Φ_b(x^a)}{Φ_b(x)}, & \text{if…
Random Guy
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For any integer $n$ there exists a prime $p$ such that the group $Z_p^*$ contains an element of order $n$

Prove that for any integer $n$ there exists a prime $p$ such that the group $\mathbb Z_p^*$ contains an element of order $n$. Show that this is possible only if $p \equiv 1\pmod{n}$. And how can I prove that for fixed $n$ there are infinitely many…
Wanksta
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Solution to $(x+1)^M-2=0\mod(x^a-1,M)$ with $a|M$.

My exercise sheet requires me to look at the following. I am looking for an elementary solution to (all polynomials have integer coefficients): Do there exist cases of a>1 and M integers with a dividing M such that $(x+1)^M-2=0\mod(x^a-1,M)$? So far…
user229769
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