Mathematics

Questions tagged [cyclotomic-polynomials]

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For questions related to cyclotomic polynomials and their properties.

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Factoring $x^{2020} + x^{2019} + \cdots + x + 1$ on $\mathbb Q[x]$ [duplicate]

An instructor asked me to factor $x^{2020} + x^{2019} + \cdots + x + 1$ on $\mathbb Q[x]$, which he considers to be tricky. This polynomial is trivial to factor on $\mathbb C[x]$ and $\mathbb R[x]$. ...
3
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1answer
50 views

Unit roots group is isomorphic to $\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right]$ in a field of characteristic $p\ge0$

Let $K$ be a field so that the group of all unit roots of all orders $\mu_\infty=\bigcup_n {\mu_n}$ (where $\mu_n=\{x\in K\mid x^n=1\}$) splits on $K$. If $K$ is of characteristic $0$, take $p=1$; ...
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1answer
21 views

Irreducibilty Cyclotomic polynomial $f(x^{n})$

Let $p$ be a prime. Let $f(x)=x^{p-1} + x^{p-2}+...+1$. Let $g(x)$ = $f(x^{n})$ where n is any positive integer. I know $f(x)$ is irreducible by Eisenstein's criterion. Now i want to show $g(x)$ ...
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1answer
24 views

Cyclotomic polynomial for any positive integer

i have searched about it but i couldn't get a clear simpler definition of this polynomial, so i need to understand more about it because i am asked how to construct it? And is this polynomial ...
2
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2answers
60 views

How can I show that $1+x+…+x^{n-1}$ divides any polynomial with distinct exponents mod n?

Given a polynomial $f(x) = x^{a_1}+x^{a_2}+...+x^{a_n}$, where each $a_j\equiv j - 1$ mod $n$, how can I show that $1+x+...+x^{n-1}$ divides $f(x)$? So far, I've noted that any root of unity $\zeta_n$...
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31 views

How can i show that the sum $\sum^{p-1}_{i=0}(-2)^{iq}$ is not a prime number?

I need to show that $\sum^{p-1}_{i=0}(-2)^{iq}$ is not a prime number, where p is a prime number greater than $5$ and $q$ a number such as $p \nmid q$. As a try, i thought that this stands for every $...
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1answer
84 views

Cyclotomic Polynomials: Example

Let be $\varepsilon$ one 9th primitive root of 1. I have to calculate Irr($\varepsilon$,$\mathbb{Q}$); this is easy, because Irr($\varepsilon$,$\mathbb{Q}$)=$\Phi_{9}(x)=x^{6}+x^{3}+1$. My problem ...
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1answer
48 views

The degree of the irreducible factors of $r^{th}$ cyclotomic polynomial over a finite field.

I'm having trouble verifying the following proposition after Lemma 4.6 in the paper PRIMES is in P: Let $Q_r(X)$ be the $r^{th}$ cyclotomic polynomial over $\mathbb{F}_p$. The Polynomial $Q_r(X)$ ...
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1answer
40 views

Prove that $(1 - \omega)^{\phi(m)} = p$ where $\omega = e^{2\pi i/m}$

I was reading Marcus' Number Fields and came across a line saying "we know $(1 - \omega)^{\phi(m)} = p\mathbb{Z}[\omega]$" where $\omega = e^{2\pi i/m}$ and $m = p^r$. Earlier on in the book it ...
2
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1answer
33 views

Quadratic extension $\mathbb{Q}(\sqrt{5})$ inside cyclotomic field $\mathbb{Q}(\zeta_{20})$

Primavera stated a question, and mentioned that... There are only three quadratic extensions over $\mathbb{Q}$ in $\mathbb{Q}(\zeta_{20})$ as $\mathbb{Q}(\sqrt{5}),\mathbb{Q}(i),\mathbb{Q}(\sqrt{5}...
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58 views

Algebraic Integers of a Field

Suppose my ring is $\mathbb{Z}[e^{i2\pi/p^2}]$. I wish to prove that the algebraic integers are ${\sum_{n = 0}^{p^2-p-1}a_{n}e^{i2n\pi/p^2}}$ where each $a_{k} \in \mathbb{Z}$. I see why thus finite ...
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1answer
32 views

Divisibility of Coefficient Related to Cyclotomic Integer

In this blog, I found the following lemma- Lemma 4: Coefficients of corresponding powers of $(Ī± - 1)$ must be congruent mod $Ī»$ provided all powers are less than the $(Ī» - 1)$ st. if: $a_0 + a_1(Ī± ...
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1answer
45 views

Solving the sextic equation with 14th root of unity

I am solving the sextic equation $t^6-t^5+t^4-t^3+t^2-t+1=0$ satisfied by the 14th root of unity (a problem from Ian Stewart's book). I was able to get up to the point where you have the polynomial $u^...
1
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1answer
49 views

Why is $x-\zeta^a$ a linear factor if $x-\zeta^p$ is a linear factor of a polynomial?

While reading a proof of why $\Phi_{n}(X)$, the $n^{th}$ cyclotomic polynomial, is irreducible in $\mathbb{Q}[X]$, I encountered a problem understanding a statement that was made: If $\zeta$ is a ...
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1answer
94 views

Trouble Proving Cyclotomic Polynomial Identity

We are required to show that $$x^n-1=\prod_{d|n}\phi_{d}(x)$$ I am aware this is considered a trivial identity and that there are numerous ways to prove it, however, I am having trouble understanding ...
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1answer
48 views

Cyclotomic Polynomial Recurrence Identity

If p is prime, show that $\Phi_{p}(x^{p^{k-1}})=\Phi_{p^{k}}(x)$. I have the solution, however I cannot make sense of how the author got to it. I understand the need to use the fact that p^k and n ...
2
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1answer
39 views

subfields of the cyclotomic fields $\mathbb{Q}(\zeta_p)$ with $p \equiv 1 [5]$

Let $p\equiv 1 (mod 5)$ a prime number, let $\mathbb{Q}(\zeta_p)$ the cyclotomic field of degree $p-1$ over $\mathbb{Q}$, its known that $\mathbb{Q}(\zeta_p)$ has unique subfield of degree $5$. My ...
2
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1answer
60 views

In Lemma 4.7 of AKS, why do distinct polynomials map to distinct elements modulo h(x)?

I am reading Primes is in P by Agrawal, Kayal and Saxena, and I can't understand part of the proof of Lemma 4.7 (already the subject of two questions here: PRIMES in P paper - Lemma 4.7 - why are the ...
2
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1answer
58 views

Divisors of $x^n+1$ over $\mathbb F_2$.

Let $\mathbb F_2=\{0,1\}$ be finite field with two elements. Are we guarantied that for all $n\in \mathbb N$ the polynomial $x^n+1$ has a divisor $g(x)\in \mathbb F_2[x]$ with the property that $g(x)...
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0answers
56 views

Minimum polynomial of the $11^{\text{th}}$ root of unity

Ring of polynomials = $\mathbb{Q}[x]$ I'm having a confusion with the theory of minimum polynomials. Say is $z$ is an $11^{th}$ root of unity. (That is: $z^{11}=1$). And I know because of the ...
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2answers
76 views

Showing that the fifteenth cyclotomic polynomial is ireducible over $\mathbb{Q}$

I need to show that $\Phi_{15}(x) = x^8 - x^7 + x^5 - x^4 + x^3 -x + 1 = \prod_{i \in \mathbb{Z}_{15}^{\times}}(x - \omega^i)$ is irreducible over $\mathbb{Q}$, where $\omega = e^{\frac{2\pi i}{15}}$. ...
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1answer
38 views

Degree of the field extension generated by $\sin(2\pi/n)$ for any natural number $n$.

I have been asked to determine the degree of the field extension over $\mathbb{Q}$ generated by $\sin(2\pi/n)$ for any natural number $n$. I have been able to show that the degree of extension by $\...
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2answers
61 views

how can find cyclotomic Polynomial where n=105

question number (D) in picture where n=105 my steps : factor 105 to 3 * 5 * 7 and can't continue to next steps
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2answers
50 views

how solve Cyclotomic Polynomial , n=8

how i can access to red box ${ \frac{x^8-1}{x^4-1} }$ ?? and how complete my solution to end result ?
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1answer
34 views

$\Phi_{2^n}(x) = x^{2^{n-1}}+1$ by Mobius Inversion

I want to prove that $\Phi_{2^n}(x) = x^{2^{n-1}}+1$. This is not hard to do by using recursion. However, I want to know if there's a simpler way to do this using the Mobius Inversion formula, which ...
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24 views

Is there a product expression for non-primitive roots of Fibonacci numbers?

I found a product related to Fibonacci numbers, from T.M. Apostol 1976. $\phi_{n}=\prod_{d|n}F_{d}^{\mu(n/d)}$, for all divisors d of n. Since the last iteration above is merely the whole Fibonacci ...
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1answer
54 views

How many prime factors will $\Phi_n$ have in its prime factorization?

Let $p$ be a prime number with $p \not |\ n.$ Let $q=p^e.$ Let $\Bbb F_q^{(n)}$ be the splitting field of $X^n-1$ over $\Bbb F_q.$ Let $\Phi_n$ denote the $n$-th cyclotomic polynomial over $\Bbb F_q.$...
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1answer
45 views

What will be $\text {Ord}_n\ q$?

I am studying Galois theory from NPTEL lecture series on finite fields and Galois theory. While watching a lecture on cyclotomic polynomial I came across a theorem which I failed to understand ...
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0answers
68 views

How does the galois group behaves with polynomial w/ an example??

Let $G$ be a multiplicative group modulo 16, i.e, $(\mathbb{Z}/16\mathbb{Z})^\times$. So we know $$ G = \{1,3,5,7,9,11,13,15\}. $$ Now, we have a cyclic group $\langle 3 \rangle = \{1, 3, 9, 11\}$ as ...
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31 views

Problem in finding $p$-th cyclotomic polynomial.

I am now studying Galois theory. Here I came across a polynomial called a cyclotomic polynomial. Let me describe it briefly. Let $K$ be the base field such that $n.1_K \neq 0$ in $K.$ Now consider ...
0
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1answer
49 views

prove cyclotomic polynomial is a minimal polynomial

I want to prove that the cyclotomic polynomial $Ī¦_n(x)$ is the minimal polynomial of the primitive $n$th root Ī¶ of unity. Is it enough to show that Ī¦ is irreducible and in Z[x]? Or is there a ...
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0answers
44 views

How to prove that if $p\mid n$, then $\Phi_{pn}(x)=\Phi_n(x^p)$?

I'm trying to solve exercise 7.5.11 from Lovett's "Abstract Algebra": Suppose that $p\mid n$. Prove that $\Phi_{pn}(x)=\Phi_n(x^p)$. Here, $\Phi_n(x)$ is the $n$th cyclotomic polynimial and I don'...
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0answers
26 views

Question about the proof of the irreducibility of cyclotomic polynomials in Dummit & Foote

In section 13.6, there is this proof: My question is about the highlighted portions. It is to my understanding that $\mathbb{Z}[x]$ is not an ED, and we cannot do division here. But how did the ...
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0answers
64 views

Quotients in cyclotomic fields: $\mathbb{Z} [\zeta_{pqr}] /(1 - \zeta_p,1 - \zeta_q) \cong \mathbb{Z} [\zeta_{r}] /(pq) $?

Let $\zeta_{l}$ be a primitive $l$th root of unity for some integer $l$, $K = \mathbb{Q}(\zeta_l)$ the $l$th cyclotomic field with $\mathcal{O}_{K} = \mathbb{Z}[\zeta_l]$ its ring of integers, and $\...
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0answers
23 views

Show that every integer is the coefficient of some $\Phi_n(x)$

I know $\Phi_{105}(x)$ has a 2 and $\Phi_{210}(x)$ has a -2 as a coefficient
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0answers
52 views

Is this an n-th root of unity (on the regular polygon)?

I'm following a proof which uses Hadamard's inequality ( $ | det(A) | \leq \prod ||(A_i) ||_2 ) $ in order to find the maximum of : $ f : D^n \rightarrow \mathbb{R} $ $(z_1, ..., z_n ) \mapsto \...
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14 views

How can we find irreducible polynomials that splits cyclotomic polynomial mod p?

How can we find irreducible polynomials that splits cyclotomic polynomial mod p? Let us consider $n$-th cyclotomic polynomials $\Phi_n(X) = \prod_{1\leq k < n, GCD(k, n)=1}(x-e^{2i \pi \frac{k}{n}})...
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1answer
95 views

Minimal polynomial of root $\zeta_n$ over finite field $\mathbb{F}_p$ [duplicate]

Let $n>1$ a number and $\zeta_n$ the the $n$-th root. $p$ be a prime. I'm looking for strategys & theorems which allow to calculate the minimal poplynomial $m_{\zeta_n}$ of $\zeta_n$ over $\...
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3answers
56 views

When does $f(x^n) \mid (f(x))^n$? And related questions.

My question: Let $f(x)$ be a polynomial with Gaussian integer coefficients. Under what conditions does $f(x^n) \mid (f(x))^n$? Can we generate all such polynomials given $d$, the degree of $f(x)$? My ...
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0answers
32 views

Irreducibility of the polynomial [duplicate]

Prove or disprove. Let $f_n(x)=x^{n-1}+x^{n-2}+\cdots +x+1$. Then $f_p(x^{p^{e-1}})$ is irreducible in $\mathbb Q[x]$ for all prime $p$. I know if $p$ is a prime $f_p(x)$ is $p$-th cyclotomic ...
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1answer
34 views

Inequality involving cyclotomic polynomial

Let $\alpha>1$, $p$ a prime number and $\phi_{k}(X)$ the $k$-th cyclotomic polynomial. I want to prove that $$ \phi_{k}(\alpha^p)-\phi_{k}(\alpha)\neq 0 $$ My attempt Still I do not have any ...
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0answers
39 views

A calculation on cyclotomic polynomial

Let $a$ be an integer such that $a$ is not the $d$-th power of an integer, for $d$ a divisor of $n$, $a>1$ and let $\alpha$ the unique positive real $n-th$ root of $a$. Supppose that $(m,n)=d$, ...
3
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1answer
134 views

Find Galois group of $x^3+x^2-2x-1\in\mathbb{Q}[x]$

Find Galois group of $f(x)=x^3+x^2-2x-1\in\mathbb{Q}[x]$ over $\mathbb{Q}$. This is not a duplication of this question because in the other question they are given a further data. My attempt: Let $...
3
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4answers
289 views

About the number of real roots

I want to solve this equation: $$t^{10}-t^{9}+ t^{8}- t^{7}+ t^{6}- t^{5}+ t^{4}- t^{3}+ t^{2}- t+1=0$$ with respect to $t$. But I have not a good idea to start. Hence, I am asking about the number ...
6
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1answer
204 views

Irreducibility of prime degree cyclotomic polynomials

I came across this proof that the cyclotomic polynomials of prime degree are irreducible over the rationals. I was wondering if anyone has come across this particular proof before. Let $p$ be prime, ...
0
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1answer
58 views

How to find the number of non-negative/positive integer solutions to $m_1x_1+\cdots+m_rx_r=n$ systematically? (coin-exchange problem)

I have been thinking about this problem for a few days already, out of curiosity. The number of non-negative integer solutions to $m_1x_1+\cdots+m_rx_r=n$ is equal to the $n$-th coefficient of $$f(z)...
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1answer
32 views

Convergence of infinite product of cyclotomic polynomials

From dabbling with p-adics, I remember mentions of $\infty $ being the 'infinite prime'. So hence something like: $\frac{x^\infty -1}{x - 1} = \displaystyle \prod_{\substack{d| p_\infty \\ d \neq 1}} \...
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1answer
53 views

Is $[\mathbb Q(\sqrt[7]2):\mathbb Q]=7$ and $[\mathbb Q(w):\mathbb Q]=6$?

Let $[\mathbb Q(\sqrt[7]2):\mathbb Q]=a$ and $[\mathbb Q(w=e^{2\pi i/7}):\mathbb Q]=b$, then 1) $a=b$ 2) $a<b$ 3) $a>b$ $[\mathbb Q(\sqrt[7]2):\mathbb Q]=7$ as $x^7 - 2$ is an irreducible ...
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1answer
65 views

Computation in cyclotomic field $\mathbb{Q}(\zeta_{7})$ over $\mathbb{Q}$.

I have some question about computation in cyclotomic field $K=\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $7$th root of unity. I know that the subfield $E=\mathbb{Q}(\zeta+\zeta^{2}+\zeta^{4})$ ...
2
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1answer
86 views

when is the $n$-th cyclotomic polynomial irreducible over $\mathbb{R}$

It is well known that the cyclotomic polynomials $\Phi_n(x)$ are irreducible over the field of rationals $\mathbb{Q}$. I am curious about their reducibility over the real numbers $\mathbb R$. We have ...

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