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Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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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 $E=\mathbb{Q}(\zeta+\zeta^{2}+\zeta^{4})$ ...
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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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Cyclotomic polynomials of distinct index are distinct.

Does there exist two cyclotomic polynomial $\Phi_n$ and $\Phi_m$ which are equal but $n\neq m$? The cyclotomic polynomial is defined as $\Phi_n(x)=\prod_{\substack{1\le j\le n \\ \gcd(j,n)=1}}(x-u_{(...
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1answer
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Factorization of $x^p-1$ modulo $p^n$

What are the (monic) divisors of the polynomial $x^p-1$ in the ring $(\mathbb{Z}/p^n\mathbb{Z})[x]$? For $n = 1$, the ring $(\mathbb{Z}/p\mathbb{Z})[x]$ is a UFD, and we have $x^p - 1 = (x-1)^p$. ...
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least degree divisor of $x^p -1$ in $Z_{q}[x]$

For p a prime and q a prime power (q not a power of p), I'm trying to find the polynomial divisors of $x^p -1$ in $Z_{q}[x]$. In particular, I'm hoping to present a general idea of the minimum degree ...
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If a monic rational polynomial of degree $p-1$ has $p$-th root of unity as a root, is it the cyclotomic polynomial?

If a monic rational polynomial of degree $p-1$ has a $p$-th root of unity as a root, where $p$ is prime, does that make it the cyclotomic polynomial $x^{p-1}+...+1$? I think this is the same as ...
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Cyclotomic cosets modulo

I've got some troubles solving an (easy ?) exercise: Let n=26, q=3. Now I'd like to find the cyclotomic cosets modulo n over $F_q$. The cyclotomic cosets are defined as $C_s = (s, ns, n^2s, ..., n^{...
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Construction of a field with roots of unity in MAGMA

I would like to construct a field (NumberField) that contains a (primitive) $n$-th root of unity and $i = \sqrt{-1}$ using a computer algebra system MAGMA, i.e. $\mathbb{Q}(\zeta_n, i)$. I tried ...
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Cyclotomic polynomial identity proof with primes.

If p is prime, show that $\Phi_{p}(x^{p^{k-1}})=\Phi_{p^{k}}(x)$. Here is my attempt: $x^{p^{k}}-1=\prod_{d|p^{k}}\Phi_{d}(x)=\prod_{i=1}^{k}\Phi_{p^{i}}(x)=\Phi_{p^{k}}(x)(\Phi_{p}\Phi_{p^{2}}\...
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Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
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How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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Quadratic polynomial satisfied by $\zeta_5+\zeta_5^{-1}$

I got one problem from Dummit Foote stating that determine the quadratic polynomial satisfied by the period $\alpha=\zeta_5+\zeta_5^{-1}$ of the the $5th$ root of unity $\zeta_5$. Determine the ...
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1answer
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Computing primitive roots of unity in a Finite Field extension

We have an irreducible polynomial $x^2 - 2 \in \mathbb{F}_5[x]$, and I have to find the primitive $12^{\text{th}}$ roots of unity in $\mathbb{F}_{5^2}$ and then compute their minimal polynomials over $...
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Hardy and Wright, Section 5.8 Clarification (construction of regular 17-gon).

As a note, this question requires some familiarity with the classic book referenced in the title, "Introduction to the Theory of Numbers", by Hardy and Wright. I have the Sixth Edition. I'm trying to ...
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Characteristic polynomial of Companion matrix $r\times r$ [duplicate]

I want to show that $a(x) = \det[C(a)-x I]$ where $a(x)=a_0+ax+...+a_{r-1}x^{r-1}+x^r$ and $C(a)$ is the companion matrix: $$\begin{vmatrix} 0&1&0&\dots& 0 \\ 0&0&1&\dots&...
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1answer
37 views

Divisors of a cyclotomic polynomial

I am not entirely sure of the validity of this source (https://proofwiki.org/wiki/Prime_Divisors_of_Cyclotomic_Polynomials), but it claims that all prime divisors $d$ of $\Phi_n(a)$ have $d\equiv 1\...
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Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$

i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$. ...
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minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$

How to find the minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$ ? And the minimal polynomial of $\sin (2\pi/11)$ over $\mathbb Q(\sqrt {11})$ ? I know that the minimal polynomial of $...
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1answer
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$g^{0p^nk}+g^{p^nk}+g^{2p^nk}+…+g^{(p-2)p^nk} \equiv 0$ (mod $p^n)$ if $p-1$ doesn't divide $k$

Let $p$ be prime and $n\geq 1$. Let $g$ be the integer equivalent of some generator for $(\mathbb{Z}/p\mathbb{Z})^ \times$. Let $k\in \{0,1,2,...\} \subseteq \mathbb{Z}$ such that $p-1$ does NOT ...
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$a \equiv b$ (mod $p$) implies $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?

Let $p$ be a prime number. If $a \equiv b$ (mod $p$), does that imply $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)? I think the answer will be yes, and I suspect that the way of proving it will involve ...
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Interpretation of Coefficients of Expanded Cyclotomic Polynomials

Working out the following definition of the Cyclotomic Polynomial $$ {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),} $$ you'll ...
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1answer
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Unique subfield of order $2$ of $\mathbb{Q(\zeta_{7}})$ over $\mathbb{Q}$

Actualy this subfield is $\mathbb{Q}(i\sqrt{7})$ since $X^{2}+7$ is irreducible over $\mathbb{Q}$ and has $i\sqrt{7}$ as root. My problem here is to show unicity, I tried something using the tower ...
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computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions

Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions. If these have been computed before, I'm looking for a ...
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1answer
54 views

$p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$

Prove that $p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$ Here $\Phi_n(x)$ is nth cyclotomic polynomial. I don't know what I should use. $$\Phi_n(x)=\prod_{{1\leq a\leq n } \& {(a,n)=1}}(x-\zeta_n^a) $$ or ...
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Question about Davenport's cyclotomy result.

In my Analytic Number Theory course, we are going through Davenport's Multiplicative Number Theory (3rd ed.), and I am having some trouble working through a certain part of section 3 (Cyclotomy). I ...
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1answer
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Proving that $x^{2^n} + 1$ is irreducible in $Q[x]$

I've been working on this and this is my process: I would like to use Eisenstein's criterion so I considered the substitution $y=x-1$. So $$x^{2^n}+1=(y+1)^{2^n}+1=\sum_{k=0}^{2^n}{2^n \choose k}y^{2^...
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1answer
81 views

Proof or reference to this property of cyclotomic polynomials

Let $n$ be a positive integer, $p$ a prime and $e \geq 1$. Denote by $\Phi_n(x)$ the $n$-th cyclotomic polynomial. Then $$\Phi_{p^en}(x) \equiv \, \Phi_n(x)^d \mathrm{mod}\,{(p)}$$ where $$d = ...
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Why a certain integral domain is not a UFD.

Let $$\mathbb{Z}[q]^{\mathbb{N}} = \varprojlim_j \mathbb{Z}[q]/((1-q)\cdots (1- q^j))$$ Why isn't $\mathbb{Z}[q]^{\mathbb{N}}$ a unique factorization domain? The author proposes a proof whose ...
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1answer
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Cyclotomic polynomial formula: is it valid in an arbitrary field?

Let $F$ be a field and $n \in \mathbb{N}$. Then an element $\varepsilon \in \overline{F}$, being $\overline{F}$ an algebraic closure of $F$, is called a $n$-th root of unity if it is a root of the ...
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Is $\mathbb{Z}_p[\omega]$ profinite when $\omega$ is a primitive $p$-root of unity?

Let $\mathbb{Z}_p$ be the $p$-adic integers. We have that the $p$-th cyclotomic polynomial is irreducible over $\mathbb{Z}_p$ applying the Eisenstein criterion (which is valid over $\mathbb{Z}_p$ when ...
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Minimal polynomial generators of aurifeuillan factors cyclotomic polynomials

In order to answer this question, I came up with a way to generate the reciprocals of aurifeuillan factors of $n$-th cyclotomic polynomials for odd prime $n$. If $n=1\pmod 4$, then $\Phi_n(nx^2)$ has ...
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Closed formula for $\Phi'_n(\zeta)$ as complex number

This is a follow-up to this question. The one thing that did not get a completely satisfactory answer there is: If $\Phi_n$ denotes the $n$-th cyclotomic polynomial, and $\zeta^k_n = e^{2k\pi i/n}$ is ...
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1answer
67 views

Relationship between cyclotomic polynomials

Let $n, m$ be two natural numbers and $\Phi_n(q), \Phi_m(q)$ the $n$-th and $m$-th cyclotomic polynomials respectively. Define a function $c_{n,m} \colon \mathbb{N} \to \left\{0,1\right\}\cup \left\{ ...
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Cyclotomic Fields are equal?

We know that the $m$th cyclotomic number field is given by $\mathbb{Q}(\xi_m)$, where $\xi_m$ is an $m$th primitive root of unity. We know also that if $\omega_m$ is another primitive root of unity, ...
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1answer
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Galois group of weird polynomial

Find $\text{Gal}(f)$ of $f=x^{2018}+1\in\mathbb{Q}(X)$. The roots are: $x=\pm i\zeta_{4036}^k$ with $k=0, 1, ..., 1009$ so at least $\Phi_{4036}\vert x^{2018}+1$ where $\deg({\Phi_{4036}})=2016$ by ...
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Question on a property related to cyclotomic fields and thaine's theorem

I am reading the part of Washington's book related to Thaine's theorem. Let $m$ be a positive integer and let $F$ be the maximal real subfield of $\mathbb Q(\zeta_m)$, i.e. $F=\mathbb Q(\zeta_m+\...
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Root of unity belongs to Z/qZ. How?

EDIT: Really sorry for not posting this initially.. maybe it's easier to understand now. Source, page 6. I've stubled upon a statement similar to this: "Let $m,q$ be two integers such that $\mathbb{...
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1answer
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On factoring polynomials whose only coefficients are 0 and 1.

I say a polynomial $P\left(z\right)=\sum_{n=0}^{d}a_{n}z^{n}$ is digital if for each $n$, $a_{n}\in\left\{ 0,1\right\}$. Let $\alpha$ be a positive integer $\geq2$, and let $P\left(z\right)$ be a ...
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Question about the Cyclotomic polynomial and Cyclotomic field.

I am studying for my Algebra PHD Qualifying and I can not solve the first 3 parts of this problem. I think I got the 4 correct. I have some ideas but I can't put them together to solve it. Anyone can ...
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Is there a nice general formula for $\displaystyle \int \frac{dx}{x^n-1}$ and/or $\displaystyle \int \frac{dx}{\Phi_n(x)}$?

Question is in the title, where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. Motivation: I'm just teaching my calculus students basic integration of rational functions with $\log$ and $\arctan$,...
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What are the intermediate fields of $\mathbb{Q}(\sqrt[4]{2},i)/\mathbb{Q}$ of order $4$ over $\mathbb{Q}$?

Let $K = \mathbb{Q}(\sqrt[4]{2},i)$. Am I correct to say that $K$ has a 8-th primitive root: $\zeta_8 = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i$? The 8-th cyclotomic polynomial is $\Omega_8 = X^4+1$ ...
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Reducibility of Cyclotomic polynomials over integers

Cyclotomic Polynomials $\Phi_n(x)$ are irreducible themselves, but when restricted to certain integers (in terms of $x$), may be reducible. The most obvious case is when $x$ is a perfect power (simple ...
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Galois group over extension field of quotient ring based on cyclotomic polynomial

Let $\Phi_m(X)$ be the $m$-th cyclotomic polynomial, $p$ be a prime, and $\mathbb{Z}_p[X]/\Phi_m(X)$ be a quotient ring. I believe that $\mathcal{Gal}(\mathbb{Q}(\zeta)\mathbb{Q}) \cong (\mathbb{Z}/m\...
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1answer
138 views

Product of all primitive nth roots of unity is 1? [duplicate]

Is the product of all primitive nth roots of unity equal to 1? Equivalently, is $\Phi_n(0) = 1$ for $n>2$? More exactly, I'm trying to prove that $$x^{\phi(n)} \Phi_n(x^{-1}) = \Phi_n(x).$$ Since ...
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1answer
76 views

Do you know a generalization of a formula about cyclotomic polynomial?

Let n be a natural number. Then n-th cyclotomic polynomial is defined as follows: $$\Phi_{n}(x)=\prod_{k\in\mathbb{N}_{<k},(n,k)=1}(x-\zeta^k)$$ where $\mathbb{N}_{<k}$ means the set of natural ...
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1answer
117 views

If a cyclotomic polynomial is reducible over a finite field, what does its factorisation look like?

The $n$th cyclotomic polynomial remains irreducible when reduced modulo $p$ if and only if $p$ is a generator of $\mathbb{Z}_n^\times$. Suppose that is not the case, and I know that the polynomial can ...
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2answers
29 views

Question about proof of Lemma in Niven's book about irrational numbers concerning minimal polynomial of a primitive nth root of unity

On page 35 in Niven's book on irrationals , during the proof of Lemma 3.6: That if $\omega$ is a primitive $n$th root of unity, and $f(x)$ is it's minimal polynomial, then for any prime $p$ such that $...
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3answers
178 views

Does there exist a formula for product of the primitive $ n $th roots of unity.

I know there is a formula for the sum of the primitive $ n $th roots of unity which is the Mobius function of $ n $. See: The Möbius function is the sum of the primitive $n$th roots of unity. I ...
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2answers
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How to program formula involving cyclotomic polynomials and Lambert series?

I want to know how to program this formula (https://people.math.gatech.edu/~mschmidt34/images/sum-of-divisors-exact-formula.png) but I can't understand the math behind or the several variables used ...
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0answers
69 views

Show the Galois group of $f(x)=x^p - 2$ is isomorphic to $\mathbb{Z}_p$

I'm having some difficulty with a problem. Here is the problem statement. Let $p$ be a prime, and $G$ be the Galois group of $f(x) = x^p - 2$ over $E$ where $E = \mathbb{Q}(\xi)$ and $\xi$ a ...