For questions related to cyclotomic polynomials and their properties.

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6
votes
3answers
108 views

Splitting $\Phi_{15}$ in irreducible factors over $\mathbb{F}_7$

I have to split $\Phi_{15}$ in irreducible factors over the field $\mathbb{F}_7$. It has been a while that I did this kind of stuff, and to be the honest, I've never really understood this matter. I'd ...
1
vote
1answer
36 views

Number of terms containing primitive root of unity

It is well known that the degree of the n-th cyclotomic polynomial is $\varphi(n)$, where $\varphi$ is the Euler totient function. I define the ${minimal}$ sum to be of the form \begin{align} \xi_0 ...
3
votes
0answers
34 views

Detect cyclotomic polynomials

I was reading this question: When does a polynomial divide $x^k - 1$ for some $k$? I followed the procedure given by Bill Dubuque in his answer (the "Graeffe" method) for the polynomial $f(x) = x+1$. ...
6
votes
3answers
175 views

Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
1
vote
1answer
37 views

Cyclotomic field of $5$ th root of Unity

Question is : Let $K$ denote the field $\mathbb{Q}(\zeta)$ where $\zeta=e^{\frac{2\pi i}{5}}$ Find $[K:\mathbb{Q}]$ Show that the splitting field of $x^{10}-1$ over $\mathbb{Q}$ is $K$ Find the ...
3
votes
1answer
42 views

Factoring Cyclotomic Polynomials Over $\mathbb{F}_p$.

How can I show that the irreducible factors of the cyclotomic polynomial $\Phi_{p^d-1}(x)$ all have degree $d$ over $\mathbb{F}_p[x]$? I'm particularly interested in a proof using the fact that for ...
2
votes
3answers
141 views

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
-1
votes
2answers
26 views

Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
9
votes
0answers
173 views

When is $(x^n-1)/(x-1)$ a prime number?

Let $x > 1$ and let $n$ be a prime. I'm wondering if a characterization of this is known. That is, what are sufficient and necessary conditions for $$ \dfrac{x^n-1}{x-1} = 1 + x + x^2 + \cdots + ...
0
votes
2answers
31 views

Primitive element of the fixed field of a subgroup of the galois group of a prime cyclotomic extension

This is a question on one step in a proof from Dummit and Foote pg. 597. Let $p$ be an odd prime (there isn't much to say for the $p=2$ case) and let $G=\text{Gal}(\mathbb{Q(\zeta_p)}/\mathbb{Q})$ ...
1
vote
1answer
47 views

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 ...
1
vote
0answers
24 views

Did I Do This Galois Theory Problem Right? Subfields of $\mathbb{Q}(\zeta_{12})$.

Let $\omega$ be a primitive $12$th root of unity. (i) What is $[ \mathbb{Q}(\omega) : \mathbb{Q}]?$ (ii) List the distinct conjugates of $\omega + \omega^{-1}$. (iii) What is $Aut(\mathbb{Q}(\omega ...
1
vote
0answers
26 views

Intermediate field of $\mathbb{Q}(\zeta_{17})/\mathbb{Q}$ by $\sigma^8$

$\zeta = \zeta_{17}$. As stated, the set up is looking at $Gal(\mathbb{Q}(\zeta_{17})/\mathbb{Q}) \simeq \mathbb{Z}/16\mathbb{Z},$ generated by $\sigma: \zeta \to \zeta^2$. I'm looking for the ...
1
vote
0answers
46 views

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
1
vote
0answers
26 views

$\zeta$ primitive $n$th root of unity, help showing that $\sqrt{n},\sqrt{-n}\in \mathbb{Q}(\zeta)$ under some conditions.

Consider $\zeta$ a primitive $n$th root of unity, show that if $n\equiv 1\mod{4}\implies \sqrt{n}\in \mathbb{Q}(\zeta)$ if $n\equiv -1\mod{4}\implies \sqrt{-n}\in \mathbb{Q}(\zeta)$. I know that ...
2
votes
1answer
63 views

Show that $\mathbb{Q}(\zeta)$ contains one of the two numbers $\sqrt{\pm5}$ and decide which one is contained in $\mathbb{Q}(\zeta)$.

Let $\zeta$ be the 15th primitive root of unity in $\mathbb{C}$, show that $\mathbb{Q}(\zeta)$ contains one of the two numbers $\sqrt{\pm5}$ and decide which one is contained in $\mathbb{Q}(\zeta)$. ...
0
votes
1answer
22 views

Lemma about a prime ideal in a commutative ring with identity

I am trying to prove the Cyclotomic polynomial is irreducible over $\mathbb{Q}[x]$ for any prime $p$ using Eisenstein's Criterion. However, I would like to be more specific and prove the following ...
2
votes
1answer
56 views

Why are two statements about a polynomial equivalent?

I am reading a claim that the following two statements are equivalent. One of the roots of a polynomial $v(t)$ is a $2^j$-th root of unity, for some $j$. The polynomial $v(t)$ is divisible either by ...
3
votes
2answers
99 views

Cyclotomic polynomial in $\mathbb{Z}/(p)$

Let $p$ be prime, $n \in \mathbb{N}$ and $p \nmid n$. $\Phi_n$ is the $n$-th cyclotomic polynomial. How can I find the maximum $n \in \mathbb{N}$ (with $p \nmid n)$ so that $\Phi_n$ splits into ...
1
vote
1answer
42 views

Nontrival Subgroups of Cyclotomic Fields

In Dummit and Foote, section 14.5, p.597, he considers the generators of $\mathbb{Q}(\zeta_{13})$ which corresponds to the subgroups of $(\mathbb{Z}/13\mathbb{Z})^{\times}\cong ...
1
vote
1answer
69 views

Roots of unity in a residue field of a Cyclotomic extension

Neukirch makes the following assertion in Algebraic Number Theory: Let $L = \mathbb{Q}(\zeta)$ where $\zeta$ is a primitve $n$th root of unity. Let $p$ be an integer coprime to $n$. For any prime ...
2
votes
3answers
78 views

Is is true that $\zeta$ has finite order?

Let $\zeta$ be a complex number on the unit circle $\{z\in \mathbb{C}: |z|=1\}$.Suppose that $[\mathbb{Q}(\zeta):\mathbb{Q}] < \infty$.Is it true that $\zeta ^n=1$ for some positive integer $n$?
0
votes
0answers
36 views

Isomorphism betwixt a Galois Group and the set of $n$th roots of unity

Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...
2
votes
1answer
55 views

Roots of Unity: Sums, Products, and Field Extensions

(1) I have to prove the following: $\forall p \in \mathbb{P} \setminus \left\{ {2}\right\}, \sum_{k=1}^{p-1}(\zeta_{p}^{k})=-1$ where $\mathbb{P}=\left\{ {p\in \mathbb{Z}:p>0, prime}\right\}$. ...
4
votes
2answers
55 views

Irreducibility of $X^n-a$

Let ${\mathbb K}$ be a subfield of ${\mathbb C}$. Let $a\in{\mathbb K}$ such that $X^d-a$ has no root in ${\mathbb K}$, for any divisor $d>1$ of $n$. Does it follow that $X^n-a$ is irreducible ...
9
votes
1answer
162 views

Cyclotomic polynomial - coefficient

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that ...
1
vote
0answers
37 views

Modulo factoring of cyclotomic polynomials

The excercise is to factorize $\Phi_7$ into irreducible factors modulo $2$. A theorem in my course: Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $n\in\mathbb{N}$ such that ...
0
votes
0answers
36 views

The cyclotomic polynomial $g_m$ is irreducible over $\mathbb{Z}_p$ if and only if $U(\mathbb{Z}_m)$ is a cyclic group generated by $\bar{p}$.

Let $p$ be a prime which does not divide $m$ and $\alpha$ be a primitive $m^{th}$ root of unity over $\mathbb{Z}_p$. Let $k$ be the order of $\bar{p}=p (\mod m)$ in the group of units in ...
2
votes
0answers
47 views

irreducible polynomial of $\alpha$ over $\mathbb{Q}$ and $[\mathbb{Q}(\alpha):\mathbb{Q}]$.

Let $\epsilon=\exp({\frac{2i\pi}{9}})$ and $\mathbb{Q}(\epsilon)$ be a cyclotomic extension of $\mathbb{Q}$ of order 9. (i) Sketch the entire lattice diagram of subfields of $\mathbb{Q}(\epsilon)$. ...
3
votes
3answers
104 views

Roots of unity modulo $p$

Let $f(X)$ be the minimal polynomial of something like $\zeta + \frac{1}{\zeta}$, where $\zeta$ is a primitive $p$-th root of unity for some prime $p > 2$. I'd like to show that $f(X) \equiv ...
3
votes
1answer
51 views

Identity Involving Cyclotomic Polynomials

Let $m > 1$ and let $p$ be a prime not dividing $m$. If $\Phi_*$ denotes that $*$th cyclotomic polynomial, then establish the following identity: $$\Phi_{pm}(x) = \frac{\Phi_m(x^p)}{\Phi_m(x)}$$ ...
1
vote
0answers
149 views

Intermediate fields of cyclotomic field $\mathbb{Q}(\zeta_8)$ - Dummit Foote $14.5.2$

Question is to : Determine the Subfields of $\mathbb{Q}(\zeta_8)$ generated by the periods of $\zeta_8$ and in particular show that not every subfield has such a period as primitive element. ...
2
votes
0answers
59 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
1
vote
1answer
62 views

Roots of unity over $\mathbb{Q}$

I want to show the following proposition from Algebra, Hungerford V.8.9. If $n > 2$ and $\xi$ is a primitive $n$th root of unity over $\mathbb{Q}$, then $[\mathbb{Q}(\xi + \xi^{-1}) : ...
8
votes
1answer
95 views

Range for values of cyclotomic polynomials, where $x$ is replaced by the golden ratio $0.61…$ ? And is it dense?

This is a recreational math question. I just played with the cyclotomic polynomials; and replacing $x$ by $1$,$-1$,$I$ gives some interesting patterns; setting $x=2$ seems to give some ...
1
vote
1answer
33 views

Proving that $|\Phi_n(x)| > x-1$

Let $\Phi_n$ be the n-th cyclotomic polynomial. I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$ The result is clear when $n$ is prime, but I'm struggling ...
2
votes
1answer
169 views

Beating Gauss on Irreducibility of Cyclotomic Polynomials?

I am considering how to provide an alternative proof of the lemma used in the proof that $\Phi_n$ is irreducible: Lemma: If $\Phi_n=f_1 f_2\cdots f_r$ is a factorization into monic irreducible ...
7
votes
0answers
125 views

Product of the first n cyclotomic polynomials.

Let $$F_n(\alpha) = \prod_{k = 1}^n \Phi_k(e(\alpha))$$ where $e(\alpha) = e^{2\pi i\alpha}$ It is clear that $F_n(\alpha) = 0$ iff $\alpha = \frac{a}{q}$ for relatively prime $a, q$ s.t. $q \le n$. ...
1
vote
1answer
175 views

How can I compute the polynomial generator for BCH?

For instance, let $C$ the binary BCH code of length $n=31$ and designed distance with auxiliary finite field $F_{32}=F_2[X]/(X^5 + X^2 + 1)$. First, we compute the cyclonitomic clases ...
0
votes
2answers
199 views

How can I factor the polynomial $x^7-1$ in GF(2)?

The result is $(x+1)(x^3+x+1)(x^3+x^2+1)$, but I don't understand how I can calculate it.
9
votes
0answers
252 views

What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...
11
votes
2answers
202 views

Derivatives of the nth cyclotomic polynomial

Are there any useful properties of the $k$th derivative of the $n$th cyclotomic polynomial? In particular, what would the value of this be at $1$ and $0$, or any properties of the value of the $k$th ...
1
vote
1answer
65 views

Computation of a certain integral involving cyclotomics

How would one compute $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz$ in terms of k and n. If this is not possible, how would someone find a good approximation for this. ...
3
votes
1answer
49 views

irreducibility of $m$th cyclotomic polynomial in $\mathbb Z[x]$ implies irreducibility in $\mathbb Q[x]$? (Ireland and Rosen)

In Chapter 13 of Ireland and Rosen's An Introduction to Classical Modern Algebra, they prove that the $m$th cyclotomic polynomial is irreducible in $\mathbb Z[x]$. Immediately afterwards they state a ...
3
votes
1answer
39 views

Program for generating the coefficients of the nth cyclotomic polynomial

Is there a program that generates the coefficients of the nth cyclotomic polynomial?
0
votes
1answer
37 views

Does the succesion of two radical extensions yield a radical extension only in the obvious case?

Answering that recent stackoverflow question, I encountered the following related problem : Let $n,m,p\geq 2$ be integers, and let $K$ be a subfield of $\mathbb C$ containing all $nmp$-th roots of ...
1
vote
0answers
256 views

Find the set of cyclotomic cosets of q modulo n

Calculating finite field and factoring $x^n - 1$ over $GF(q)$ first step is to calculate cyclotomic cosets. For example : For $n=9,q=2$ $C_1=\{1,2,4,8,7,5\} = C_4 = C_8 = C_7 = C_5$ $C_3=\{3,6\} ...
1
vote
0answers
42 views

Divisors and cyclotomic polynomials

Let $n \in \mathbb{N}^{\ast}$ and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial defined by : $$ \Phi_{n}(X) = \prod \limits_{\substack{1 \leq k \leq n-1 \\ \gcd(k,n)=1}} \Big( X - \exp \big( ...
3
votes
1answer
92 views

Cyclotomic Polynomial Evaluated at 1

I noticed that when trying to evaluate the $n$th cyclotomic polynomial at $1$, $\Phi_n(1)$, we run into an issue while using the explicit formula $\Phi_n(x)=\prod_{d|n}(x^d-1)^{\mu(n/d)}$. In ...
3
votes
3answers
270 views

Cyclotomic polynomial

I want to prove basic results of Cyclotomic polynomial over $\mathbb{Q}$ $(1) \Phi_n(0)=1$ for all $n>1$ $(2) \Phi_{2n}(x)=\Phi_n(-x)$ for all odd number $n>1$ I want use this result ...