For questions related to cyclotomic polynomials and their properties.

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11
votes
1answer
82 views

Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
0
votes
4answers
52 views

determining which cyclotomic polynomial is $x^8 -x^4+1$

Given that the following polynomial\begin{equation*}f(x)=x^8-x^4+1\end{equation*} is a cyclotomic polynomial $\Phi_n$ for some $n\in \mathbb N$. are there some basic tools to determine $n$? I know ...
1
vote
1answer
31 views

$p$-adic Euler's totient function

Let $p$ be a prime number, $\overline{\mathbb{Q}_p}$ a fixed algebraic closure of $\mathbb{Q}_p$. Let $\alpha$ be a primitive $n$-th root of unity in $\overline{\mathbb{Q}_p}$ and $d$ its degree over ...
0
votes
0answers
34 views

A question about the degree of sin(2π/n) over the rationals

In cyclotomic theory, $\cos (2\pi /n)$) is shown to have degree $\varphi (n)/2$ over the rationals $\mathbb{Q}$, while $\sin (2\pi /n)$has degree $\varphi (n)$ (as long as n is not divisible by ...
8
votes
2answers
64 views

Irreducibility of Cyclotomic polynomials over number field

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at most ...
1
vote
1answer
61 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
0
votes
1answer
30 views

Prove that $Φ_{nm}(x) = Φ_n(x^m)$ if every prime divisor of m is also divisor of n

Let $m$ and $n$ be natural numbers that every prime divisor of $m$ is also a divisor of $n$. We can define $Φ_{ab}(x)$ for every prime $a>0$ like this: $$Φ_{ab}(x) = \begin{cases} Φ_b(x^a), & ...
0
votes
1answer
29 views

Find a polynomial in $\mathbb{Z}_{41}$

Find a $7^{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}$, so that $$ p(14^i) = i\ (mod\ 41)\ \forall i = 0,1,\ldots,7. $$ $3$ is the $8^{th}$ primitive root of unity and $3 * 14 = 8 * 36 = 1$ ...
1
vote
1answer
32 views

Is there a relationship between these polynomial concepts?

I'm currently doing a bit of reading on abstract algebra (more specifically Polynomial Theory), and noticed something that may have some sort of significance perhaps? The section I'm reading on at ...
2
votes
2answers
160 views

A unit of seventh cyclotomic field

I have troubles with the following problem about units. Show that $1+\zeta $, $1+\zeta+\zeta^2$ are units in the field $\mathbb{Q[\zeta]}$, where $\zeta$ is a seventh primitive root of unit ...
0
votes
0answers
14 views

Do the minimal cyclotomic polynomials always have coefficient 1, 0 or -1?

I noticed the following trend in the minimal cyclotomic polynomials for each $\zeta_{k}$. $\zeta_{5}\rightarrow t^4+t^3+t^2+1\\\zeta_{6}\rightarrow t^2-t+1\\\zeta_{7}\rightarrow ...
1
vote
1answer
40 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...
0
votes
2answers
58 views

For any integer $n$ there exists a prime $p$ such that the group $Z_p^*$ contains an element of order $n$

Prove that for any integer $n$ there exists a prime $p$ such that the group $\mathbb Z_p^*$ contains an element of order $n$. Show that this is possible only if $p \equiv 1\pmod{n}$. And how can I ...
0
votes
1answer
17 views

Primitive roots of unity proof verification

"Let $C_n(x)$ be the polynomial such that the roots of $C_n(x)=0$ are the primitive $n^{th}$ roots of unity. Prove that there are no positive integers $q,r,s$ for which $C_q(x)=C_r(x)C_s(x)$." My ...
3
votes
3answers
55 views

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
90 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
38 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 + ...
2
votes
2answers
43 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
48 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
35 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
votes
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
49 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
45 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
votes
0answers
22 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. ...
1
vote
0answers
56 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
vote
0answers
37 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
votes
2answers
34 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
35 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
69 views

Intermediate fields 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
106 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
44 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
votes
1answer
70 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 ...
1
vote
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 ...
1
vote
0answers
42 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 ...
7
votes
1answer
209 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
79 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
65 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
377 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 ...
14
votes
0answers
224 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
82 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
93 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 ...
1
vote
0answers
55 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
votes
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
votes
2answers
92 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 ...
8
votes
0answers
200 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
2answers
117 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
136 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 ...