# Questions tagged [cyclotomic-polynomials]

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

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 ...
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### 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 ...
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### 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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### 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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### $\Phi_{2^n}(x) = x^{2^{n-1}}+1$ 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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### 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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### 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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### 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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### 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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### 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 ...
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### prove cyclotomic polynomial is a minimal polynomial

I want to prove that the cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of the primitive $n$th root ζ of unity. Is it enough to show that Φ is irreducible and in Z[x]? Or is there a ...
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### 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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### 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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### 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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### 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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### 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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### 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 $(m,n)=d$, ...
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### Is $[\mathbb Q(\sqrt2):\mathbb Q]=7$ and $[\mathbb Q(w):\mathbb Q]=6$?

Let $[\mathbb Q(\sqrt2):\mathbb Q]=a$ and $[\mathbb Q(w=e^{2\pi i/7}):\mathbb Q]=b$, then 1) $a=b$ 2) $a<b$ 3) $a>b$ $[\mathbb Q(\sqrt2):\mathbb Q]=7$ as $x^7 - 2$ is an irreducible ...
### 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 $E=\mathbb{Q}(\zeta+\zeta^{2}+\zeta^{4})$ ...
### 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 ...