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Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

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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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1answer
75 views

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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1answer
24 views

$a \equiv b$ (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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elementary question concerning cyclotomic polynomials

Is there a way to prove by induction (I can do it purely using the definition and a bit of playing around) that a polynomial in the form $z^{2^n}-1$ has a cyclotomic polynomial in the form $z^{2^{n-1}}...
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2answers
69 views

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
28 views

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
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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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1answer
54 views

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
55 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
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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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2answers
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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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1answer
65 views

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
48 views

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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1answer
82 views

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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424 views

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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1answer
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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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1answer
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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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1answer
47 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
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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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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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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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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
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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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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
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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 ...
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1answer
37 views

What does $\varphi (n)$ denote in the context 'the class of $q$ modulo $n$ has order $\varphi (n)$'?

I am currently trying to understand the following condition (found here in Corollary 48) which tells us when the $n^\text{th}$ cyclotomic polynomial $\Phi_n (x)$ (over a finite field) is irreducible: ...
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0answers
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Question about proof regarding cyclotomic polynomials from Dummit & Foote

I'm currently reading through Dummit & Foote's Abstract Algebra textbook. I'm in section 13.6, Cyclotomic Polynomials and Extensions, which is focused around proving that $[\mathbb{Q}(\zeta_n):\...
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34 views

Question about cyclotomic polynomials (arising from the paper “PRIMES is in P”)

Within the paper PRIMES is in P (found here), the following passage can be found on page 5: Let $Q_r(X)$ be $r^\text{th}$ cyclotomic polynomial over $F_p$. Polynomial $Q_r(X)$ divides $X^r − 1$ and ...
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1answer
28 views

Question about the uniqueness of the $n^\text{th}$ cylclotomic polynomial?

I am currently trying to understand the concept of a cyclotomic polynomial which, due to my absent knowledge of rings/fields/etc, is proving difficult. From Wikipedia, the following definition is ...
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0answers
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Question related to N-th cyclotomic polynomial, principal N-th root of unity and residue class of X

I am struggling to understand a couple of statements in a cryptography-related paper. I think I lack some maths background. Can you help me understand it ? Here are the statements: We consider the ...
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137 views

Solving $x^2+x+1=7^n$

Do simple equations $x^2+x+1=7^n$ have infinitely or finitely many solutions? In fact, what about the general equation $x^2+x+1=p^n$ where $p$ is a prime congruent to $1$ $\pmod 3$? Are there any ...