# Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

468 questions
Filter by
Sorted by
Tagged with
19 views

### 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 ...
56 views

1 vote
25 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).
27 views

### 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 ...
48 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 ...
25 views

### 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 $i$ ...
45 views

### 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 ...
1 vote
121 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 ...
59 views

1 vote
56 views

### What is the correspondent sub-field of this cyclotomic extension, given by the Fundamental Theorem of Galois Theory?

I've been working on some problems regarding polynomials, and I ended up with the following question: Suppose that $\zeta_{p^n}$ is a root of unity of order $p^n$, and consider the cyclotomic ...
1 vote
134 views

1 vote
110 views

### 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 ...
209 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 $p$. ...
54 views

### 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(\...
47 views

### 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 $s = 4m - 3$...
50 views

### 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 ...
1 vote
43 views

74 views

### 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 ...
89 views

### 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 ...
117 views

### 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 ... 371 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 ...
1 vote
75 views

### 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 ...
45 views

### 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 ...
### 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 ...
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\...