Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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On some congruence in cyclotomic field

Let $k = \mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field and $p$ a prime number verify $p\equiv 1 \pmod 5$. Let $\lambda = 1-\zeta_5$ the unique prime of $k$ above $5$, precisely $5 = (1-\zeta_5)^...
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Factoring cyclotomic polynomials in tree form

$\newcommand\ord{\text{ord}_n(q)}$ I've got the following formula for the $n$'th cyclotomic polynomial over the field $\mathbb{F}_q$ ($q=p^r$ and $(n,p)=1$) assuming $q\equiv1(\bmod m)$ as well as ...
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3 votes
2 answers
119 views

Show that $1+\zeta+\dots+\zeta^{k-1}$ is a unit [duplicate]

Let $\zeta=e^{\frac{2\pi i}{p}}$ where $p\geq3$ is a prime. Consider the algebraic number field $K=\mathbb{Q}(\zeta)$. Let $k$ be a positive integer such that $(k,p)=1$ i.e., $k$ is co-prime with $p$. ...
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Question on cyclotomic polynomials and the Möbius function

I'm doing an exercise on the Möbius function $\mu$. I've seen this equation but I don't understand it. \begin{align*} \mathrm{X}^{n}-1&= \prod_{d \mid n} \phi_d(X) \\ &=\prod_{d \mid n} \left(\...
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Irreducible polynomial $\Phi_5(x)$ over $\mathbb Z$ is not necessarily prime when considered as an element of $\mathbb F_p[x]$ [duplicate]

Say $\Phi_5(x) = \sum_{i=0}^4 x^i \in\mathbb Z[x]$ and note that $x^5 -1 = (x-1)\Phi_5(x)$. If $r(x)\in\mathbb F_3[x]$ is a monic prime factor of $\Phi_5(x)$ in the same polynomial ring, then we know ...
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Necessary and sufficient condition on prime $p$ where cyclotomic polynomial $\Phi_{15}(x)$ has certain number of distinct roots in $\mathbb F_p$

Def'n $\Phi_{15}(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 \in \mathbb Z[x]$. Further, $x^{15} -1 = (x-1)(x^2 + x + 1)(x^4 + x^3 + x^2 + x + 1)\Phi_{15}(x)$ where $x^3 -1 = (x-1)(x^2 + x + 1)$ and $x^5 ...
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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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How to write an arbitrary element of cyclotomic extension of 6th root of unity?

Let's say $\omega$ is a primitive $6^{\rm th}$ root of unity. Then we know $[\mathbb{Q}(\omega):\mathbb{Q}]=\varphi(6)=2$. If I count the number of linearly independent basis elements for $\mathbb{Q}(\...
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1 answer
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Calculate $N(1-\xi_m)$, where $\xi_m= e^{\frac{2\pi i}{m}}$ and $m=p^l$

Let $K=\mathbb{Q}[\xi_m]$ be a number field, where $\xi_m= e^{\frac{2\pi i}{m}}$ and $m=p^l$. I want to calculate the norm of $(1-\xi_m)$. I know how to do this if $l=1$, but if $l> 1$ I get a bit ...
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Simple proof that n divides $\sum_{1\le k<n, \gcd(k,n)=1} k$ [duplicate]

I was reading about cyclotomic polynomials defined as: $$ \Phi_n(x) = \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1} \left(x-e^{2i\pi\frac{k}{n}}\right)$$ The fact that the coefficients of these ...
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Particular case of the law of quadratic reciprocity

Let $\mathbb{F}_q$ be a finite field of characteristic $p \neq 2,5$ . What I've shown so far : $x \in\mathbb{F}_q^* $ is a root of $\Phi_5 = X^4 + X^3 +X^2+X+1$ if and only if $x$ is of order 5 in $\...
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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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3 votes
0 answers
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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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2 votes
4 answers
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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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3 votes
1 answer
122 views

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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  • 102
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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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1 answer
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Norm of generator of prime ideal

Let $K=\mathbb{Q}(\zeta_n)$ be a cyclotomic field. For a prime ideal $\mathfrak{p}$ in the ring of integers $\mathcal{O}_K$, we can write $\mathfrak{p} = (p,f_i(\zeta_n))$, where $\mathfrak{p}\cap\...
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  • 251
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2 answers
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For fixed positive integers $a,k$ prove that there are infinitely many primes $p$ for which $p\equiv 1\pmod{k}$ and $a$ is a $k$'th power mod $p$

For fixed positive integers $a$ and $k$, prove that there are infinitely many primes $p$ for which $p\equiv 1\pmod{k}$ and $a$ is a $k$th power modulo $p$. I know that this problem could be solved ...
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$K/\Bbb Q< \infty$ extension and a prime $p\nmid |K:\Bbb Q|$ with $\zeta_p \notin K$. Is $\Phi_p(X)$ is irreducible over $K$?

Let $K/\Bbb Q$ be a finite extension and $p$ be a prime such that $p\nmid |K:\Bbb Q|$ and $\zeta_p \notin K$. Can we say that the $p$-th cylotomic polynomial $\Phi_p(X)$ is irreducible over $K$? The ...
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2 votes
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What is the discriminant of $\Phi_{2}(X)$ over $\Bbb Q$?

What is the discriminant of $\Phi_{2^n}(X)$ over $\Bbb Q$ for $n=1$ since $\Phi_2(X)=X+1$ has only one root $-1$? I have calculated $disc(\Phi_{2^n}(X))$ for $n\geq 2$ which matches exactly with what ...
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  • 2,241
0 votes
2 answers
108 views

Prove $Q_{mp} = Q_{m}(x^p)$ when prime $p$ divides $m$. ($Q$ is cyclotomic polynomial over some finite field $K$)

All I know is playing with this fact that $Q_{mp}(x) = \dfrac{x^{mp}-1}{\prod_{d|mp}Q_d(x)}$ and I really did many calculations but they went nowhere. Also I proved that $Q_{mp}(x) = Q_m(x^p)/Q_m(x)$ ...
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  • 55
3 votes
0 answers
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Smallest $p$ such that $2\mid\operatorname{ord}(a)$ in $\mathbb{Z}/p\mathbb{Z}$

Fix a constant $a$. Then consider primes $p$ and the sequence $$a^1-1, a^2-1, a^3-1, a^4-1,...,a^N-1.$$ We say a prime $p$ is a primitive divisor of $a^m-1$ if it divides $a^m-1$ and does not divide $...
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  • 1,482
1 vote
0 answers
50 views

For $\omega=e^{2\pi i/q}$ and $c_0,\ldots,c_{q-1}\in \mathbb Z$, $\sum\limits_{k=0}^{q-1}c_k\omega^k \equiv \sum\limits_{k=0}^{q-1}c_k\pmod p$.

Let $q$ be a power of a prime $p$, and $\omega=e^{2\pi i/q}$. Suppose $c_0,\ldots,c_{q-1}\in \mathbb Z$ and that $\sum\limits_{k=0}^{q-1}c_k\omega^k \in \mathbb Z$. Then I have to show that $$\sum\...
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0 votes
1 answer
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How to prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$?

How do I prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$? I calculated some examples in Wolfram Alpha where the results were of this specific form, and it ...
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1 vote
1 answer
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Book recommendation/reference request on a gentle introduction to cyclotomic polynomials

Can anyone recommend a book or reference material (written in English) that offers a gentle introduction to cyclotomic polynomials? The book does not have to be too comprehensive. I plan to apply the ...
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1 vote
1 answer
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$p_1^x\equiv p_2^x\equiv 1 \pmod{p_1p_2-1}$ problem

Let $p_1,p_2$ be primes and $x\in\mathbb{N}$. I want to investigate \begin{equation*} p_1^x\equiv p_2^x\equiv 1 \pmod{p_1p_2-1} \end{equation*} I want to find how $x$ depends on $p_1$ and $p_2$. This ...
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1 vote
0 answers
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Prime number of the form $\phi_n(n^p)$?

Let $n\ge 2$ be an integer and $p\ge 7$ be a prime number. Is there a prime of the form $\phi_n(n^p)$ , where $\phi_n$ denotes the $n^\text{th}$ cyclotomic polynomial ? We must have $p\mid n$ ...
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Smallest prime of the form $\phi_{21209}(n)$?

What is the smallest positive integer $n$ such that $$\phi_{21209}(n)=\frac{(n^{21209}-1)(n-1)}{(n^{127}-1)(n^{167}-1)}$$ is a prime number ? I chose the $21209$ th cyclotomic polynomial because the ...
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2 votes
1 answer
75 views

Tight bounds for $\Phi_n(b)$?

What are tight bounds for $$\Phi_n(b)$$ where $\ \Phi_n()\ $ denotes the $\ n $-th cyclotomic polynomial , $\ n\ge 3\ $ and $\ b\ $ is a "large" positive integer ? We can assume that the ...
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4 votes
0 answers
44 views

Can we rule out $\ p\mid t\ $ , if $\ p\mid \phi_t(n)\ $?

Let $\ \phi_m(x)\ $ denote the $\ m\ $-th cyclotomic polynomial. For $\ n\ge 2\ $ define $\ t:=\phi_n(n)\ $ Must all the prime factors $\ p\ $ of $\ \phi_t(n)\ $ satisfy $\ p\equiv 1\mod t\ $ ? It ...
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2 votes
0 answers
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How many Carmichael numbers exist of the form $\Phi_n(b)$ for $n>6$?

Let $\Phi_n(x)$ be the $n$ - th cyclotomic polynomial. Are there Carmichael numbers of the form $\Phi_n(b)$ , where $b$ is a positive integer and $n$ is a positive integer different from the numbers $...
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6 votes
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Can $\Phi_n(\Phi_n(n))$ be a prime number?

Can $$f(n):=\Phi_n(\Phi_n(n))$$ where $n$ is a positive integer, be a prime number , where $\Phi_n(x)$ denotes the $n-th$ cyclotomic polynomial ? If $n$ is a prime power $p^k$ ($p$ prime , $k$ ...
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0 votes
2 answers
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A question related to cyclotomic polynomials in -x

Consider the following problem from Hungerford Algebra : If n is odd, then $g_{2n}(x) = g_n(-x)$ where $g_n(x) $ are cyclotomic polynomials over $\mathbb{Q}$. So, $g_{2n} (x) = \frac {x^{2n}-1} {\...
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  • 1,498
0 votes
1 answer
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Prove that the splitting field of $x^p-1$ for p an odd prime contains a unique intermediate field of degree 2

Let $p$ be an odd prime. Let $F$ be the splitting field of $x^p-1$ over $\mathbb{Q}$. Prove that there is a unique field $K$ bewtween $\mathbb{Q}$ and $F$ which is of degree 2 over $\mathbb{Q}$. I ...
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  • 369
22 votes
2 answers
438 views

Is the "cyclotomic diagonalization" always squarefree?

For every integer $n\ge 2$ , define $$f(n):=\Phi_n(n)$$ where $\Phi(n)$ is the $n$ th cyclotomic polynomial. This can be considered as the "cyclotomic diagonalization" Prove or disprove the ...
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1 vote
1 answer
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Inverses in $\mathbb{Z}[\zeta_p]$

I'm trying to solve the following exercise from my algebraic number theory class: Let $\zeta_p$ be a p-th root of unity. We want to show that for $1\leq k,j\leq p-1$, $$\frac{1-\zeta_p^k}{1-\zeta_p^j}$...
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  • 649
2 votes
1 answer
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Cyclotomic polynomial dividing $f\in\mathbb{Q}[X]$

I know that $\varphi (n)\geq \sqrt{\frac{n}{2}}$ for natural numbers $n$, and that the cyclotomic polynomials are irreducible in $\mathbb{Q}[X]$. Let $f\in\mathbb{Q}[X]$ be a polynomial of degree $m$ ...
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3 votes
1 answer
40 views

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 ...
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4 votes
0 answers
123 views

Splitting field of $8$-th root of the unity is Galois

Let $\omega=e^{i\pi/4}$ be the $8$-th root of the unity. I know that $\mathbb{Q}(\omega)/\mathbb{Q}$ is Galois, as it is the splitting field of the separable polynomial $p(x)=x^8-1$. However, I ran ...
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Dot Product of Vectors of Roots of Unity

Let $n \in \mathbb{N}$ and $i$ be the usual complex number such that $i^2=-1$. Let $\zeta = \exp(\frac{\pi}{n} i)$, $v = [1,\zeta,\zeta^2,\ldots,\zeta^{n-1}]$. Given $J \subseteq \{1,\ldots,n\}$, let $...
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0 answers
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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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1 vote
0 answers
158 views

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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5 votes
2 answers
119 views

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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2 votes
1 answer
115 views

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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  • 2,509
1 vote
0 answers
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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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  • 35
0 votes
1 answer
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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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  • 35
0 votes
1 answer
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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..
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