For questions related to cyclotomic polynomials and their properties.

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2
votes
3answers
71 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
25 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
30 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
41 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 ...
10
votes
1answer
111 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
28 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
25 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
41 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
70 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
39 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
97 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
38 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
53 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
85 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
30 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
128 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
115 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
75 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
79 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
183 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
176 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
63 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
46 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
34 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
36 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
118 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
37 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
75 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
180 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 ...
4
votes
2answers
289 views

When is a cyclotomic polynomial over a finite field a minimal polynomial?

When is the cyclotomic polynomial $f(x)$ over a finite field $\mathrm{F}_q$ also the minimal polynomial of some element $\alpha \in \mathrm{F}_q$?
8
votes
4answers
597 views

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
2
votes
1answer
67 views

Cyclotomic field automorphisms “fill up” $\mathbb{Z}/n\mathbb{Z}$?

I know from reading that the Galois group of a cyclotomic polynomial is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$. While I believe this, I can't figure out why that should work. In particular, it's ...
1
vote
0answers
124 views

Factorization of Cyclotomic Polynomials mod p

Suppose $p$ is a prime such that $p\nmid n$, and let $\alpha$ be a root of $\overline{\Phi}_n(x)$ in some extension field of $\mathbb{Z}_p$, where $\overline{\Phi}_n(x)\in\mathbb{Z}_p[x]$ denotes the ...
2
votes
0answers
93 views

Cyclic linear codes and idempotents

Got this assignment from coding class and would be very thankful for checking if my solutions are correct. a) Find all idempotents modulo $1 + x^{17}$ of degree at most $15$ So first i find $r$ from ...
0
votes
1answer
63 views

On the ring of integers of cyclotomic fields

I'm doing an exercise whose main purpose is to show that $\mathscr{O}_K=\mathbb{Z}[\xi]$, where $K=\mathbb{Q}[\xi]$ and $\xi$ is a primitive $p$-th root of $1$, $p$ prime. So, let ...
1
vote
1answer
81 views

$k[x]/(x^n-1)$ is isomorphic to…?

I am wondering if I can relate the quotient of $k[x]$ by the relation $x^n-1=0$, where $k$ is a field of characteristic zero, to roots of unity.
4
votes
1answer
101 views

Irreducibility of Cyclotomic Polynomial over a Cyclotomic Extension

I am trying to prove that $\Phi_m(x)$ is irreducible over $\Bbb Q(\zeta_n)$ if and only if $(m,n)\leq2$. The left implication turns out to be somewhat easy since without loss of generality, $2\mid m$ ...
2
votes
1answer
83 views

Calculating $\frac{\Phi_p(x^n)}{\Phi_p(x)}$ using generating functions or inclusion-exclusion

Where$$\Phi_p(x^n)=1+x^n+\cdots+x^{n(p-1)}$$ is the $p$th cyclotomic polynomial in $x^n$, and $p$ is prime. For instance, ...
4
votes
2answers
239 views

Determine the degree of the splitting field for $f(x)=x^{15}-1$.

Let $f(x)= x^{15} - 1$. Let $L$ be the splitting field of $f(x)$ over the field $K$. Determine the extension degree $[L:K]$ in each case. a) $K= \Bbb{R}$ b) $K= \Bbb{Q}$ c) $K = ...
6
votes
1answer
144 views

$\mathbb{Q}(\sqrt[3]{2})$ is not in any cyclotomic extension of $\mathbb{Q}$ [duplicate]

Question is to prove that $\mathbb{Q}(\sqrt[3]{2})$ is not in any cyclotomic extension of $\mathbb{Q}$. As i was not so sure how to proceed i did the following thing(which may possibly be not so ...
4
votes
3answers
268 views

Minimal Polynomial of $\zeta+\zeta^{-1}$

Question is to find Minimal Polynomial of $\zeta+\zeta^{-1}\in \mathbb{Q}(\zeta)$ over $\mathbb{Q}$ Where $\zeta$ is primitive $13^{th}$ root of Unity. What all i know is that Minimal polynomial of ...
0
votes
1answer
128 views

Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$

It seems I have forgot all the details about the group theory. Anyone knows what is the Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$ and is that solvable? Thanks.
3
votes
2answers
124 views

McDaniel's restriction on odd perfect numbers and use of cyclotomic polynomials

In 1969, McDaniel showed that an odd number of the form $$N=p^\alpha\prod_{j=1}^tp_j^{2\beta_j},$$ where $p$, $p_1$, $p_2$, $\ldots$, $p_t$ are distinct primes and $$p\equiv\alpha\equiv1\pmod4,$$ ...
11
votes
0answers
216 views

Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that ...
5
votes
1answer
46 views

The minimal polynimial of a primitive $p^{m}$-th root of unity over $\mathbb{Q}_p$

Proposition 7.13 of Neukirch's ANT states that for a primitive $p^{m}$-th root of unity $\zeta$ (p prime) the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is totally ramified of degree ...
2
votes
3answers
98 views

Why $x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$?

Why $f(x) = x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$ (where $\zeta = \zeta_7$ is a primitive root of the unit) over $\mathbb{Q}$? Of course it's irreducible by the ...
0
votes
1answer
40 views

When $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$?

maybe this is a stupid question . Anyway, when $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$, i.e., $\phi(n) = 2$ (where $\phi$ is the Euler's totient function)? Thanks in advance.
4
votes
1answer
105 views

What are the bounds on the class number of a cyclotomic field with regulator power of 2?

Let $\mathbb{Q}(\zeta_n)$ be the $n$th cyclotomic field with $n$ being a power of $2$. What is the best known asymptotic upper bound on the class number of $\mathbb{Q}(\zeta_n)$ as n grows? Can we ...
3
votes
0answers
80 views

Gaussian periods

Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the ...
0
votes
1answer
38 views

Formula for the $nm$th cyclotomic polynomial when $(n,m) = 1$

Let $n,m$ be coprime. I want to find a formulae for $\Phi_{n\cdot m, \mathbb Q}$. I conjecture that because $$d \mid nm \implies d \mid n \lor d \mid m,$$ that $$ \Phi_{n\cdot m, \mathbb Q} = ...