For questions related to cyclotomic polynomials and their properties.

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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 ...
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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 ...
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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 ...
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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 $\phi$ function which gives the number of coprime elements. ...
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$\zeta$ 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 ...
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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 $\mathbb{Q}(\zeta)$. ...
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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 ...
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1answer
55 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 ...
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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 ...
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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 ...
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3answers
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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 $n$?
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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 $\mathbb{P}=\left\{ {p\in \mathbb{Z}:p>0, prime}\right\}$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
158 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 ...
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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 ...
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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 $\mathbb{Q}(\epsilon)$. ...
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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 ...
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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: $$\Phi_{pm}(x) = \frac{\Phi_m(x^p)}{\Phi_m(x)}$$ ...
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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. ...
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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 ...
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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. $q \le n$. ...
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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}) : ...
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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 ...
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1answer
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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 ...
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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, ...
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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 ...
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134 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.
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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, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...
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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. ...
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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 ...
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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 ...
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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 ...
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Program for generating the coefficients of the nth cyclotomic polynomial

Is there a program that generates the coefficients of the nth cyclotomic polynomial?
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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\} ...
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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( ...
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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 ...
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1answer
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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 ...
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Series of Cyclotomic polynomials

How can I show that if $\Phi$ is a Cyclotomic polynomial, $$\Phi_n(x)=\prod_{1\leq k\leq n}_{(n,k)=1}(x-\zeta_n^k)$$ With $\frac{d}{dx}\Phi_n(x)=\Phi'_n(x)$ Then, ...
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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 $\alpha \in \mathrm{F}_q$?
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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 ...
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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 ...
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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 ...
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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 ...
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Cyclotomic Polynomial of a Prime

I have this question on a homework sheet: Claim:$$\Phi_{p}(x)=1+x+x^2+...+x^{p-1}\space$$ for $p$ prime. which was followed by the claim that $\Phi_{p^n}(x)=\Phi_p(x^{p^{n-1}})$ which I have ...
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1answer
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$k[x]/(x^n-1)$ 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.
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1answer
104 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, $2\mid m$ ...
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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 ...