For questions related to cyclotomic polynomials and their properties.

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3
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3answers
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for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$ Note: $\Phi_n(x)$ is the $n$th cyclotomic polynomial whose roots are the primitive $n$th roots of unity if n is odd then $-1$ cannot ...
3
votes
0answers
79 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
1
vote
1answer
34 views

Irreducible polynomials in $\Bbb F_{2}[x]$ and their generators.

Hi am trying to find as many irreducible polynomials in $\Bbb F_{2}[x]$ as possible and their generating cyclotomic cosets. So far I have found 3 - a number am quite pleased with; $1 + x + x^2 + x^3 + ...
1
vote
1answer
39 views

Binomial of Mersenne prime power.

Let $f(x)$ be irreducible over $\mathbb{Z}_2$ of degree $p$, where $p$ is prime. Let $2^p-1$ be a Mersenne prime. I have to show \begin{equation*} f(x) \mid (x^{2^p-1}-1). \end{equation*} I am ...
2
votes
2answers
38 views

Let $\zeta=e^{2\pi i/n}$ Prove that $x^n -1 =(x-1)(x-\zeta)(x-\zeta^2) \dots (x-\zeta^{n-1})$

This is a question about cyclotomic polynomials and I have already shown that $x^n-1 =\Pi\Phi_d(x)$, taking the product over all divisors d of n.
2
votes
3answers
44 views

Roots of “almost cyclotomic” polynomials

While looking at the behavior of the probability of getting a run of $k$ heads during $n$ tosses of a fair coin, I ended up needing to know the nature of the roots of polynomials of the form $$ P(x) = ...
0
votes
1answer
29 views

Q basis for splitting field

I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it ...
0
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0answers
33 views

Solution to $(x+1)^M-2=0\mod(x^a-1,M)$ with $a|M$.

My exercise sheet requires me to look at the following. I am looking for an elementary solution to (all polynomials have integer coefficients): Do there exist cases of a>1 and M integers with a ...
1
vote
1answer
48 views

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$.

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$. How to derive this. I am having no clue. I guess $n|m$. But what is the proof?
0
votes
1answer
40 views

Let $G$ be the finite abelian group show that there is a galois extension $K/\Bbb Q$ with $Gal(K/\Bbb Q) \equiv G$ [duplicate]

Let $G$ be the finite abelian group show that there is a galois extension $K/ \Bbb Q$ with $Gal(K/\Bbb Q) \cong G$. I have seen one proof using For a fixed positive integer $n$, there are infinitely ...
0
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0answers
20 views

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$ [duplicate]

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$. I have thaught that for $\Bbb Q(\sqrt 3)$ is in $\Bbb Q(w)$ where $w$ is in the $3$rd root of unity. ...
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0answers
54 views

Cyclotomic polynomial, after adjoining a radical

Suppose that $p>2$ is prime and that $a$ is a rational number for which $\sqrt[p]{a}$ is in $\mathbb C\backslash\mathbb Q$. The cyclotomic polynomial $\Phi_p$ is well-known to be irreducible over ...
1
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0answers
34 views

Lucas's Cyclotomic Formula

There are 2 well-known formulas involving Cyclotomic polynomials, which can be described roughly as writing $\phi_n$ as norms of elements in some quadratic extension. They appear in wikipedia and ...
0
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2answers
33 views

Why does the automorphism mapping $\omega$ to 1, not an element of Galois group

Let $L/\mathbb{Q}$ be a field extension, where $L=\mathbb{Q}(\omega)$ and $\omega=e^{\frac{2\pi i}{7}}$. In my textbook it states that $Aut(L/\mathbb{Q})=\{\sigma_i|\sigma_i(\omega)=\omega^i, 1\leq ...
1
vote
1answer
32 views

Irreducibility of cyclotomic polynomials of prime order

I am stuck with an exercise, where I have to prove the $\Phi_5[X] \in \mathbb{F}_2(x)$ is irreducible. I know that $X^5-1=\Phi_5(X)(X-1)$ (shown in previous part of the exercise) $X^2+X+1$ is the ...
4
votes
1answer
57 views

Intermediate fieds for a Cyclotomic Polynomial of order $27$?

I would like to determine the Galois structure of the field $K=\Bbb Q(\zeta_{27})$--the rationals adjoined a primitive $27^{th}$ root of unity. That is to say I would like to determine the ...
2
votes
2answers
95 views

Stuck when tackling the computation of $\Phi_n(\zeta_8)$

My current way of calculation of $\Phi_n(\zeta_8)$ where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial and $\zeta_8=\cos(\frac{2\pi}{8})+i\sin(\frac{2\pi}{8})$ leave me now stuck at the problem of ...
2
votes
2answers
40 views

Why do the cyclotomic cosets have the same size?

So I have this problem: when $p=2$ and $n=2^m-1$ show that $|C_1|=|C_3|=m$ where $m\geq 3$. $C_s=\{s,ps,p^2s,\dots p^{m_s-1}\}$ is the cyclotomic coset. I know that the size of the coset is $m_s$ ...
5
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1answer
63 views

Cyclotomic fields: Determining the fixed field

Let $K=\mathbb Q$ and $L=\mathbb Q(\zeta_8)$, where $\zeta_8$ is a primitve $8th$ root of unity. I have to determine the Galois group of this extension, the subgroups of it and the associated ...
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0answers
37 views

Bunyakovsky conjecture for cyclotomic polynomials

This article on Wikipedia: http://en.wikipedia.org/wiki/Bunyakovsky_conjecture says: In fact, it can be shown that if for all natural number $ n $, there exists a natural number $ x > 1 $ such ...
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0answers
34 views

quartic subfield of cyclotomic field

I want to know which is the quartic subfield of the cyclotomic field $\mathbb{Q}(\zeta_p)$ where $p$ is an odd prime? it is $\mathbb{Q}(\sqrt{\varepsilon\sqrt{p}})$ where $\varepsilon$ is the ...
5
votes
1answer
183 views

Cyclotomic polynomials and Galois group

Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity. Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$. I already know ...
2
votes
2answers
78 views

An $n \times n$ matrix with rational entries such that $A^{n+1}=I$

I'm working on finding $A \in M_n(\mathbb{Q})$ such that $A^{n+1}=I$. If $n$ is odd, $A=-I$ satisfies the condition. When $n$ is even, clearly it should have eigenvalues $e^{2 \pi ik/(n+1)}(k=1,\cdots ...
2
votes
0answers
26 views

Show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime. [duplicate]

How to show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime where $\Phi_n(X)$ is the cyclotomic polynomial define by ...
2
votes
2answers
62 views

Polynomial with positive coefficients

Consider a polynomial $P(x) = \sum_{i=1}^{n}{a_ix^{i-1}}$ in $\mathbb{C}$. Is it true that if $\{a_i\}$ are positive and not all equal, then $P(\exp(\frac{2i\pi}{n})) \neq 0$ ? Thanks
39
votes
1answer
370 views

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
13
votes
0answers
207 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
3
votes
2answers
78 views

If $\zeta= e^{2\pi i/7}$ then $2\cos(2\pi/7) = (\zeta+ \zeta^6)$. Find a cubic polynomial satisfied by $\zeta + \zeta^6$.

My approach was to set $\zeta + \zeta^6 = a_1x + a_2x^2 + a_3x^3$ and then solve for the constants. I know there is a better way to solve this using the cyclotomic polynomials, but I don't understand ...
7
votes
1answer
88 views

how can I find the smallest integer $n$ such that a polynomial divides $x^{n}-1$

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example ...
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0answers
52 views

elementary properties of cyclotomic polynomials

How can one rewrite $1+x^2+x^4+x^8+\cdots x^{2^n}$ as a product of cyclotomic polynomials? more general how can we express $1+x^p+\cdots+x^{p^n}$, where $p$ is a prime, in term of product of ...
0
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0answers
25 views

Cyclotomic polynomial identity [duplicate]

STATEMENT Show that $$\Phi_n(X)=\prod_{d\mid n}(X^{n/d}-1)^{\mu(d)}$$ where $\Phi_n(X)$ is the n-th cyclotomic polynomial. QUESTION Could anyone offer any help on how to show this result.
2
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2answers
91 views

For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$

Well I attempted to try this but I failed to solve it: So $$\Phi_{p^{n}}(x)= \frac{x^{p^{n}} - 1}{\Phi_{1}(x) \Phi_{p^{2}}(x) \dots \Phi_{p^{n-1}}(x)}$$ Now I'm just stuck here. I saw a result ...
0
votes
0answers
33 views

Points on a unit circle

Let $P_1, P_2,..., P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2,3,...,2013\}$ is the product of all pairwise distances: $$\prod_{1\le i\lt j\le n} P_{i}P_{j}$$ a ...
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0answers
183 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
3
votes
1answer
97 views

Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
2
votes
1answer
22 views

Find a positive integer $n$ such that there is a subfield of $\mathbb{Q}_n$ that is not a cyclotomic extension of $\mathbb{Q}$

I'm in trouble with this simple exercise. If we denote $\mathbb{Q}_n:=\mathbb{Q}(\omega)$, where $\omega$ is a primitive $n$th of unity in $\mathbb{C}$, can we find a positive integer $n$ such that ...
3
votes
4answers
131 views

Norm of an element in cyclotomic extension (Exercises VI.19 Lang's Algebra)

Let $\zeta$ be a primitive $n^{\rm{th}}$ root of unity. Let $K=\mathbb{Q}(\zeta)$. If $n=p^r (r\geq 1)$ is a prime power, show that $N_{K/F}(1-\zeta)=p$ If $n$ is divisible by at least two distinct ...
4
votes
1answer
33 views

product considering the period of index over a cyclotomic extension

This is an exercise from Milne Galois Theory (Chapter 3 Exercise 13). Let p be an odd prime, and let $\zeta$ be a primitive $p^{\text{th}}$ root of 1 in $\mathbb C$. Let $E = \mathbb Q[\zeta ]$, and ...
4
votes
1answer
61 views

Inertia Degree in Cyclotomic Extensions

Let $\zeta$ be a primitive $l$th root of unity, where $l$ is prime. If $p$ is another prime number, let $f$ be the order of $p$ in $U(\mathbb{Z}/l \mathbb{Z})$. Then in $\mathbb{Z}[\zeta]$, $p$ ...
5
votes
1answer
97 views

A cyclotomic polynomial whose index has a large prime divisor cannot be too sparse

Working on this recent MSE question, I was led to the following conjecture : Suppose that $n$ is an integer with at least one prime divisor $\geq 7$. Then $\Phi_n$ has at least seven non-zero ...
0
votes
0answers
51 views

Factorization of seventh cyclotomic polynomial

The fifth cyclotomic polynomial $\Phi_5(z)$ factors as $$ \Phi_5(z)=(z^2+\varphi z+1)(z^2+(1-\varphi)z+1) $$ where $\varphi$ and $1-\varphi$ are the solutions to $x^2-x-1$. Of course, $\varphi$ is ...
2
votes
2answers
54 views

When is a sum of consecutive roots of unity an integer

Let $\xi \neq 1$ be an $n$th root of unity. When is a sum of the form $$ 1+\xi+\xi^2+\ldots+\xi^r, \quad 1 \leq r \leq n-1, $$ an integer? What are the possible integers? I suspect that the answers ...
0
votes
1answer
67 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
3
votes
0answers
68 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
6
votes
3answers
137 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
48 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
43 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 $f(x) = x+1$. ...
7
votes
3answers
211 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
70 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
89 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 ...