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Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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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
52 views

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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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 $...
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$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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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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$p|\Phi_n(2)$ 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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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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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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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
66 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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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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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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Relationship between $m$-th cyclotomic polynomial and multiplicative group $(Z/mZ)^*$ [closed]

I'm studying cyclotomic polynomial. What's the relationship of those?
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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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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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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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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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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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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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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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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 ...
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1answer
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irreducible polynomial over finite field.

I'm working on the following problem: Let $K$ be the splitting field of $f(x)=x^{6}+x^{3}+1$ over $\mathbb{Z}_{2}$, and let $\alpha$ be a root of $f(x)$. (1) Show that $\alpha^{9}=1$ but $\...
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For which $m$ does a primitive $m^{th}$ root of unity have degree $2$ over $\mathbb{Q}$?

To be honest, I don't really know whether it is okay to post this question here since perhaps the proof is really trivial... This is the question from Lang's Algebra Chapter VI Q18 part b). It asks ...
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Repeated roots of polynomials defining cyclotomic fields $\pmod p$

Let $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Because all elements in the finite field GF($p$) are roots of the polynomial $x^p-x$, it follows that this polynomial has no repeated roots $\pmod p$...
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1answer
117 views

Generalization of $x^n-1$ following Fermat's Little theorem

Fermat's Little Theorem tells us that if $\gcd(x,p)=1$ and $p$ is prime, then $x^{p-1} = 1 \pmod p$. Equivalently, if $a(p)=x^{p-1}-1$, then $a(p-1)=0 \pmod p$ if $p$ is prime. When $p > 2$, $a(p-1)...
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Let $n \geq 3$ and let $p$ be prime. Show that $\sqrt[n]{p}$ is not contained in a cyclotomic extension of $\mathbb{Q}$ [duplicate]

I think that the following statement is correct. Let $n \geq 3$ and let $p$ be prime. Show that $\sqrt[n]{p}$ is not contained in a cyclotomic extension of $\mathbb{Q}$. My idea for a proof is ...
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2answers
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Determine Minimal Polynomial of Primitive 10th Root of Unity

I would like to determine the minimal polynomial of the primitive tenth root of unity (denoted $\zeta$) over $\mathbb{Q}$. I know that the polynomial is given by: $\prod_{\text{gcd}(a,10)=1)\text{ ...
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1answer
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Is the image of $\Phi_n(x) \in \mathbb{Z}[x]$ in $\mathbb{F}_q[x]$ still a cyclotomic polynomial?

Let $\Phi_n(x) \in \mathbb{Z}[x]$ denote the $n$-th cyclotomic polynomial, and let $\mathbb{F}_q$ be the finite field with $p^k = q$ elements ($p$ prime). Let $\Phi'_n(x)$ be the reduction of $\Phi_n(...
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2answers
55 views

Proving the identity $\Phi_{np}(x) = \Phi_n(x^p)/\Phi_n(x)$, with $p \nmid n$

I've been going through the Wikipedia article on cyclotomic polynomials, trying to prove the various identities that can be used as tools to compute $\Phi_n(x)$. One such identity that I'm not sure ...