Mathematics Stack Exchange
Mathematics Stack Exchange

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
8
votes
3 answers

Proving that cyclotomic polynomials have integer coefficients

I don't understand why Gauss's lemma is invoked in the proof in Dummit and Foote that $\Phi_n(x)$ (the $n$th cyclotomic polynomial) belongs to $\mathbb{Z}[x]$. I'm an analyst and I wanted to remind myself about cyclotomic polynomials. The following…
jordanbell2357
  • 245
6
votes
1 answer

A cyclotomic polynomial whose index has a large prime divisor cannot be too sparse

Working on this recent MSE question, I was led to the following conjecture : Suppose that $n$ is an integer with at least one prime divisor $\geq 7$. Then $\Phi_n$ has at least seven non-zero coefficients. I have checked this conjecture up to…
Ewan Delanoy
  • 61,600
6
votes
3 answers

Cyclotomic polynomial

I want to prove basic results of cyclotomic polynomials over $\mathbb{Q}$. $(1)\ \Phi_n(0)=1$ for all $n>1$. $(2)\ \Phi_{2n}(x)=\Phi_n(-x)$ for all odd numbers $n>1$. I want use this result as definition: For every positive integer $n$, $$…
Siddhant Trivedi
  • 2,298
5
votes
1 answer

Product of cyclotomic polynomials

Prove that $$\prod_{_{m \mid p_1 \cdots p_n}}\Phi_{2m}(2) = 2^{p_1p_2 \cdots p_n}+1,$$ where $p_1,\ldots,p_n$ are distinct primes. I was wondering how to use the definitions of cyclotomic polynomials in order to prove this statement. Should we…
user19405892
  • 15,592
3
votes
1 answer

Lower bound for cyclotomic polynomials evaluated at an integer

In a paper about Zsigmondy's theorem, there is the following statement as a remark, without proof, nor reference: Let $n, a $ be integers $(\gt1)$ and $q$ a prime divisor of $n=q^{i}.r$, such that $q$ does not divide $r$, let $b=a^{q^{i-1}}$, let…
René Gy
  • 3,716
3
votes
1 answer

Positive coefficients after the substitution?

Let $n>1$ be an integer and $\Phi_n(x)$ be the $n$ th cyclotomic polynomial. If we subsititute $x$ by $x+1$ , do we always get a polynomial with positive coefficients ? If yes, how can this be proven ? I also want to prove that no coeffcient upto…
Peter
  • 84,454
2
votes
1 answer

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 the same degree. I have a confusion with this…
eraldcoil
  • 3,508
2
votes
0 answers

Show that every integer is the coefficient of some $\Phi_n(x)$

I know $\Phi_{105}(x)$ has a 2 and $\Phi_{210}(x)$ has a -2 as a coefficient
Kai
  • 874
  • 4
  • 14
2
votes
1 answer

Cyclotomic polynomial formula: is it valid in an arbitrary field?

Let $F$ be a field and $n \in \mathbb{N}$. Then an element $\varepsilon \in \overline{F}$, being $\overline{F}$ an algebraic closure of $F$, is called a $n$-th root of unity if it is a root of the polynomial $X^n - 1 \in F[X]$. If I denote by $W_n$…
joseabp91
  • 2,360
1
vote
0 answers

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 $p$-th roots of $ω_i$, which must be primitive…
NotaChoice
  • 931
  • 3
  • 15
1
vote
1 answer

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..
roydiptajit
  • 738
1
vote
1 answer

Irreducibilty Cyclotomic polynomial $f(x^{n})$

Let $p$ be a prime. Let $f(x)=x^{p-1} + x^{p-2}+...+1$. Let $g(x)$ = $f(x^{n})$ where n is any positive integer. I know $f(x)$ is irreducible by Eisenstein's criterion. Now i want to show $g(x)$ is irreducible: If $g(x) = h(x)\cdot k(x) $, Then…
Mukil raj
  • 39
1
vote
2 answers

Are cyclotomic polynomials irreducible modulo a prime?

I am actually interested on cyclotomic polynomials of the form $x^n+1$ (i.e. $n$ is a power of 2, for large $n$). Are these polynomials irreducible modulo a prime $q$? I already saw that $X^2+1$ is not irreducible modulo 5 because $Z_5$ contains…
Tal-Botvinnik
  • 1,843
1
vote
1 answer

Irreducible polynomials in $\Bbb F_{2}[x]$ and their generators.

Hi am trying to find as many irreducible polynomials in $\Bbb F_{2}[x]$ as possible and their generating cyclotomic cosets. So far I have found 3 - a number am quite pleased with; $1 + x + x^2 + x^3 + x^4, 1 + x + x^4 and 1 + x^3 + x^4$. I think…
guthik
  • 429
1
vote
3 answers

Roots of "almost cyclotomic" polynomials

While looking at the behavior of the probability of getting a run of $k$ heads during $n$ tosses of a fair coin, I ended up needing to know the nature of the roots of polynomials of the form $$ P(x) = x^k - \sum_{m=0}^{k-1} x^m $$ It appears that…
Mark Fischler
  • 41,743
1
2