For questions related to cyclotomic polynomials and their properties.

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23
votes
1answer
368 views

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, ...
11
votes
2answers
187 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 ...
11
votes
0answers
235 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 ...
9
votes
1answer
141 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 ...
9
votes
4answers
946 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, ...
9
votes
0answers
229 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, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...
8
votes
1answer
88 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 ...
7
votes
2answers
186 views

Explicit Galois theory computation in cyclotomic field

Let $p$ be an odd prime. Let $\zeta=e^{\frac{\pi i}{4 p}}$, thus $\zeta$ is a primitive $8p$-th root of unity. There is a unique $\mathbb Q$-automorphism $\tau$ of the number field ${\mathbb ...
7
votes
0answers
121 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. $q \le n$. ...
6
votes
1answer
196 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 ...
5
votes
1answer
188 views

Cyclotomic Polynomials and GCD

Since Cyclotomic polynomials are irreducible over $\mathbb{Q}$, $\phi_n(x)$, $\phi_m(x)$ are coprime as polynomials in $\mathbb{Z}[x]$. Working over $\mathbb{Q}$, $(\phi_n(x)$, $\phi_m(x))=(1)$. ...
5
votes
2answers
228 views

Given a prime $p\in\mathbb{N}$, is $A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}[x]$?

If $p \in \mathbb{N}$ is a prime, is $\displaystyle A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}[x]$? I don't think it is. If somebody sees a contradiction, I would be glad to see it. ...
5
votes
1answer
49 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 ...
4
votes
3answers
338 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 ...
4
votes
2answers
317 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 = ...
4
votes
2answers
53 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 ...
4
votes
2answers
430 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 $\alpha \in \mathrm{F}_q$?
4
votes
1answer
117 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 ...
4
votes
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, $2\mid m$ ...
3
votes
3answers
250 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 ...
3
votes
1answer
233 views

Galois group of a cyclotomic extension

Do you know a condition on the field $k$ for that the injection of $Gal(k[\zeta_n]/k)$ in $(Z/nZ)^*$ is bijective ? It is the case for $k = \mathbb{Q}$, but not for $k = \mathbb{R}$ for example.
3
votes
1answer
48 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
3answers
155 views

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 ...
3
votes
1answer
37 views

Program for generating the coefficients of the nth cyclotomic polynomial

Is there a program that generates the coefficients of the nth cyclotomic polynomial?
3
votes
1answer
65 views

Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?

I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write: $ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $ So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means ...
3
votes
1answer
148 views

Cyclotomic Fields and Cyclotomic Polynomials

In lectures we defined the cyclotomic polynomials as $\Phi_n(x):=\prod_{k=1,\gcd(k,n)=1}^n (x-e^{2\pi i k/n})$ and the cyclotomic field $K_n$ as the splitting field of $x^n-1$ over $\mathbb Q$. But we ...
3
votes
2answers
91 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 ...
3
votes
3answers
90 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
49 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: $$\Phi_{pm}(x) = \frac{\Phi_m(x^p)}{\Phi_m(x)}$$ ...
3
votes
1answer
80 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
2answers
129 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 $$p\equiv\alpha\equiv1\pmod4,$$ ...
3
votes
0answers
92 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 ...
2
votes
3answers
76 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 $n$?
2
votes
3answers
103 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 ...
2
votes
3answers
73 views

Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.

Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$. What I've done so far: Let $\phi(n) = p$ so the following products each have p components. $$\Phi_{n}(x) = \prod_{k=1, ...
2
votes
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 ...
2
votes
1answer
59 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 $\mathbb{Q}(\zeta)$. ...
2
votes
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 ...
2
votes
1answer
86 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, ...
2
votes
1answer
80 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 ...
2
votes
1answer
156 views

Cyclotomic polynomials, irreducibility [duplicate]

I need to decide if certain cyclotomic polynomials are irreducibles over the $\mathbb{F}_q$. For example, if $\Phi_{12}(x)$ is irreducible over $\mathbb{F}_9$. Anyone can help me? Ok, i think i ...
2
votes
1answer
87 views

Values of cyclotomic polynomial and properties of groups

There are restrictions on the values of a cyclotomic polynomial evaluated at an integer that are reminiscent of the restrictions on the number of Sylow groups of a group. So I'd like to know if there ...
2
votes
1answer
47 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 $\mathbb{P}=\left\{ {p\in \mathbb{Z}:p>0, prime}\right\}$. ...
2
votes
0answers
46 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 $\mathbb{Q}(\epsilon)$. ...
2
votes
0answers
54 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 ...
2
votes
0answers
124 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 ...
1
vote
1answer
84 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.
1
vote
1answer
32 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 ...
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}) : ...
1
vote
1answer
41 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 ...