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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 ...
node196884's user avatar
5 votes
1 answer
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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 ...
user967210's user avatar
1 vote
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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 ...
lkksn's user avatar
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2 votes
2 answers
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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[...
JoJo P's user avatar
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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 $n > 3$ ...
lux fun's user avatar
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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 ...
roundsquare's user avatar
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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 ...
Stephanie V's user avatar
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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 $p$...
NotaChoice's user avatar
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2 answers
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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 ...
Stephanie V's user avatar
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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, ...
Raheel's user avatar
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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 $...
William Sun's user avatar
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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^...
Ty Perkins's user avatar
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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 ...
user43208's user avatar
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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 ...
Mako's user avatar
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Product of all $n$th-Cyclotomic polynomials less than $m$

Has the following function been studied? I'm looking for a reference, or an answer displaying some identities and properties $$ f_m(x) = \prod_{n=1}^{m}\Phi_n(x) $$ It looks like a factorial analog of ...
Mako's user avatar
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4 votes
1 answer
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Discriminant of numbers

In a previous post on a $q$-analog of number theory I present the fact that any $q$-number can be written as a unique product of cyclotomic polynomials, similar to the fundamental theorem of ...
Mako's user avatar
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Cyclotomic polynomial as minimal polynomial

I'm in the process of learning Galois theory and got stuck on Wikipedia's alternative definition of the $n$th cyclotomic polynomial as the "minimal polynomial over the field of the rational ...
kalanchloe's user avatar
1 vote
1 answer
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Same field extension, different irreducible polynomial?

I've come across an issue while doing an assigment for a field theory subject. The issue seems pretty basic to me but I can't figure it out it's solution by myself. Let's consider the field extension $...
Pau Vallès Martínez's user avatar
2 votes
0 answers
61 views

Is This Irreducible?

Is the following analog of the cyclotomic polynomial irreducible in $\mathbb{Q}{[x]}$? $$\Phi_n(q;\chi) = \prod_{\gcd(k,n)=1}(q-\chi_n(k)\sqrt{\zeta_{n}}^k) \quad 1\leq k \leq n$$ Where $\chi_: \left(\...
Mako's user avatar
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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) = \...
LexLarn's user avatar
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For all complex $|z| \neq 1$ : $\frac{z}{1+z+z^{2}+z^{3}+z^{4}} = \sum_{n=0}^{\infty} T_n \frac{z^n}{1+z^n+z^{2n}}$?

Inspired by this one For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$? It made sense to me, to take ...
mick's user avatar
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For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?

Ok I am a bit confused. So here comes a question, Consider a maclaurin series for $f(z)$ $$f(z) = \sum_{n=0}^{\infty} f_n z^n$$ where $f(z)$ has a radius of exactly $1$. $f(z)$ may or may not have a ...
mick's user avatar
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1 vote
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Find a prime p such that the 35th cyclotomic polynomial has an irreducible 12th-degree factor in $F_p$

I have to find a prime p such that the 35th cyclotomic polynomial has an irreducible 12th-degree factor in $F_p$. I suppose that the roots of the 35th cyclotomic polynomial have order 1,5,7 or 35. The ...
Lucía Nieto's user avatar
2 votes
1 answer
80 views

Is there an example, such that $\omega\in F(\beta)\land \omega\notin F$?

$F$ is perfect, $E=F(\beta)$, where $\beta^n=a\in F$, then $E$ is a Galois extension over $F$ if and only if the field contains the n-th primitive root $\omega$, where $\omega^n=1$ Question: I am ...
GGplay's user avatar
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Find $n$ that satisfies $\Phi_p(x)=C(x)^2+(-1)^{\frac{n+1}{2}}n\cdot D(x)^2$

Let $p>2$ be a prime number, based on Gauss's formula shows that cyclotomic polynomial $\Phi_p$ can be expressed as $$\Phi_p(x)=A(x)^2+(-1)^{\frac{p+1}{2}}p\cdot B(x)^2$$ where $A,B\in \mathbb{Q}[x]...
Fukuzawa Yukichi's user avatar
0 votes
1 answer
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Bounds for cyclotomic polynomials?

Whilst going through online forums, I found this thread post, which claims For $n\in Z >1$, $$S\cdot n^{\phi(r)} \leq \Phi_r(n)\leq \frac{1}{S}\cdot n^{\phi(r)}$$ where $ S = \prod_{m=1}^{\infty} (...
Sahaj's user avatar
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2 votes
0 answers
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An efficient algorithm for factorising cyclotomic polynomial over a finite field

I need to factor a cyclotomic polynomial $\Phi_m(X)$ modulo $\mathbb{Z}_{p^e}$ where $p$ is a relatively small (at most 8 bits) prime number, and $e$ is a positive integer. As far as I know, the ...
Lukie Boy's user avatar
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0 answers
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Primitive root of unity of irreducible factor of cyclotomic polynomial over finite field [duplicate]

Sorry if the title is not descriptive, I couldn't find a way to properly describe the problem. In the proof of the AKS algorithm (link), the following is stated in Lemma 4.7: "First note that ...
user avatar
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0 answers
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Impact of plaintext modulus overflow in (B)FV homomorphic encryption

I just tried reading the paper "Somewhat Practical Fully Homomorphic Encryption" by Fan and Vercauteren but I got confused by their use of the coefficient-wise modulus. They define the ...
dj_rydu's user avatar
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2 votes
1 answer
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Compute Intermediate Extensions of $\Bbb Q[\zeta]/\Bbb Q$.

Let $\zeta$ be a primitive $n^{\operatorname{th}}$ root of unity over $\Bbb Q$ then there is a natural isomorphism $$\operatorname{Gal}\left(\Bbb Q\left[\zeta\right]/\Bbb Q\right)\simeq \left(\Bbb Z/...
William Sun's user avatar
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0 answers
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A prime $\mathcal{P}$ dividing $\mathbb{Q}(\zeta)$ divides $\Phi_{m}(\alpha)$

I came across the proof of the following fact in the Chapter $2$, Lemma $2.9$ of Lawrence CWashington's book, Introduction to Cyclotomic Fields. Let $p$ be a rational prime, $n\in\mathbb{N}$ and $a\in\...
HumbleStudent's user avatar
3 votes
2 answers
157 views

Using cyclotomic polynomials to show a polynomial is irreducible over $\mathbb{Q}$

I have been given the polynomial $$f(x)=x^8+x^7-x^5-x^4-x^3+x+1$$ and have been asked to show it is irreducible over $\mathbb{Q}$ by considering the product $(x^2-x+1)f$. (Looking it up, I realise $f$ ...
IntegralPrime's user avatar
2 votes
0 answers
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Cyclotomic Polynomials and The Existence of Infinite Prime Power

Prove that there exist infinitely many positive integers n such that all prime divisors of $n^2 + n + 1$ are not greater than $\sqrt{n}$ This is a problem related to cyclotomic polynomial. It is ...
FaranAiki's user avatar
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1 vote
1 answer
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For $f \in \mathbb{Q}[x]$, does there necessarily exist $a \in \mathbb{Z}$ such that $af \in \mathbb{Z}[x]$ and is primitive?

I'm working through a proof from lecture notes in an algebra class that for all $n$, the $n$-th cyclotomic polynomial $\Phi_n$ has integer coefficients. The proof proceeds as follows: Write $x^n - 1 =...
Joel Newman's user avatar
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0 answers
71 views

Polynomials of the form $X^n-c$

Is there any specific name for the polynomials of the form $X^n-c$ for some $n$ and $c$ is a complex number? They seem to be closely related to Cyclotomic polynomials.
user2167741's user avatar
0 votes
2 answers
78 views

Show that $Tr(A)$ is in the spectrum of $A+I$

If we have $p$ an odd prime number and the matrix $A$ $(p×p)$ with rational entries and we know that $\det(A^{p}+I)=0$, $\det(A+I)≠0$. How we can show $\operatorname{Tr}(A)$ is in the spectrum of $A+I$...
Stefan Solomon's user avatar
1 vote
1 answer
137 views

Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$

In my notation, $\mathbb{Z}_n$ = the integers modulo $n$. Find a Galois extension whose Galois group is $\mathbb{Z}_4 \times \mathbb{Z}_4$. (The additive group). I thought about using cyclotomic ...
Nicolas Torres's user avatar
3 votes
0 answers
116 views

Intermediate Fields of $\mathbb{Q}(\zeta_{16})$ and Corresponding Splitting Polynomials

I'm trying to find all intermediate fields $\mathbb{Q} \subset E \subset \mathbb{Q}(\zeta_{16})$ and in particular, the polynomial that each of these fields split. I know that $\text{Gal}(\mathbb{Q}(\...
Dalop's user avatar
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1 answer
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Frobenius automorphism modulo power of prime

This question is related to the one I have asked yesterday. For an integer $m\geq 1$, consider the $m$th cyclotomic polynomial $\Phi_m(X)\in\mathbb Z[X]$. For an integer $r\geq 1$, we can find ...
Zuy's user avatar
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0 votes
1 answer
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Isomorphism between cyclotomic rings

This question is similar to the one posted here. Let $m=m_1\cdots m_k$ be a positive integer decomposed into coprime prime powers $m_i$. For an integer $n\geq 1$, denote the $n$-th cyclotomic ...
Zuy's user avatar
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2 votes
2 answers
154 views

Divisors count of $3^n\pm1$

During the investigation of $3^n\pm1$ visually saw that the divisors count of it are sums of $2^m$. Especially if we take $n=p_1p_2$ ($p_i$ is prime) then at least for $n=p_1p_2<130$ it is either ...
Gevorg Hmayakyan's user avatar
0 votes
4 answers
100 views

$m+1$ generates the kernel of $Z_n^\times\to Z_m^\times$ where $m\mid n$ with the same prime factors

Suppose $m\mid n$. Using the First Isomorphism Theorem with respect to the homomorphism $$\begin{array}{rccc}f:&\mathbb{Z}_n^\times&\to&\mathbb{Z}_m^\times \\&x&\mapsto &x\bmod ...
Joseph Johnston's user avatar
1 vote
0 answers
28 views

Why is $\mathbb{F}_{p^r}$ the splitting field of the polynomial $X^{p^r-1}-1$? [duplicate]

Why is $\mathbb{F}_{p^r}$ the splitting field of the polynomial $X^{p^r-1}-1$? This is mentioned on my lecture notes but I don’t have a reference for this fact (and I couldn’t find it online).
dahemar's user avatar
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Any normal third degree extension of $\mathbb Q$ contained in $\mathbb C$ must be contained in $\mathbb R$

We let $K\subset \mathbb C$ be a finite normal field extension of $\mathbb Q$ such that $[K:\mathbb Q]=3$. Show that $K\subset \mathbb R$. I know that no cyclotomic extension of $\mathbb Q$ will do ...
Daniel Cortild's user avatar
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0 answers
73 views

On cyclotomics modulo prime powers

Regarding the ring $\mathbb{Z}[X]/\Phi_n(X)$ for the $n$'th cyclotomic polynomial $\Phi_n$, I find much literature on the nature of this ring taken modulo a prime $q$, that is literature on the ...
Joseph Johnston's user avatar
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0 answers
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If $u$ is a unit of $\mathbb{Z[\xi_p]}$, $\frac{u}{\bar{u}}\neq -\xi_p^i$ for all $i$.

I am trying to prove the following: Let $p$ be a prime. If $u$ is a unit of $\mathbb{Z[\xi_p]}$ (where $\xi_p$ is a primitive $p$-th root of unity), then $\frac{u}{\bar{u}}\neq -\xi_p^i$ for all $i$ ...
ABC's user avatar
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2 answers
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If $u$ is an $n$-th root of unity, $p$ is a prime number such that $p \nmid n$, then $P(u^p)=0$ where $P$ is the minimal polynomial of $u$

This is a step in a guided proof that the cyclotomic polynomial $\Phi_n$ is the minimal polynomial of $u$. I already know that $\Phi_n(0)=0$ so $P$ divides $\Phi_n$, I need to show the converse. Any ...
lanero's user avatar
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1 vote
0 answers
314 views

Prove that the roots of cyclotomic polynomial $\Phi_{p-1}(x) \equiv 0 (mod~p)$ are exactly the primitive roots mod p

$p$ is a prime, and $\Phi_{p-1}(x)$ denote the cyclotomic polynomial of order $p-1$. And I want to show the following: $g$ is a solution of the congruence $\Phi_{p-1}(x) \equiv 0 (mod~p)$ if and only ...
Gang men's user avatar
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2 votes
1 answer
131 views

If $1,\alpha_1,\alpha_2,..,\alpha_{n-1}$ denote the $n^{th}$ roots of unity, then what is $\prod_{k=1}^{n-1}(1-\alpha_k)$

Here's what we did in class : $f(z)=z^n-1$ has all of the $n$ $n^{th}$ roots of $1$ as it's zeroes. Let's say that $\alpha_1=e^{\iota(2\pi/n)}$ and $\alpha_j=(\alpha_1)^j$ for $j\in[1,n-1]$. So $1,\...
Rajdeep Sindhu's user avatar

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