Questions tagged [cyclotomic-polynomials]
For questions related to cyclotomic polynomials and their properties.
2
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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 $...
0
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1answer
29 views
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 ...
1
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0answers
18 views
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:
$$\begin{vmatrix}
0&1&0&\dots& 0 \\
0&0&1&\dots&...
0
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1answer
34 views
Divisors of a cyclotomic polynomial
I am not entirely sure of the validity of this source (https://proofwiki.org/wiki/Prime_Divisors_of_Cyclotomic_Polynomials), but it claims that all prime divisors $d$ of $\Phi_n(a)$ have $d\equiv 1\...
0
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0answers
36 views
Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$
i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$.
...
4
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1answer
89 views
minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$
How to find the minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$ ? And the minimal polynomial of $\sin (2\pi/11)$ over $\mathbb Q(\sqrt {11})$ ?
I know that the minimal polynomial of $...
1
vote
1answer
45 views
$g^{0p^nk}+g^{p^nk}+g^{2p^nk}+…+g^{(p-2)p^nk} \equiv 0$ (mod $p^n)$ if $p-1$ doesn't divide $k$
Let $p$ be prime and $n\geq 1$. Let $g$ be the integer equivalent of some generator for $(\mathbb{Z}/p\mathbb{Z})^ \times$.
Let $k\in \{0,1,2,...\} \subseteq \mathbb{Z}$ such that $p-1$ does NOT ...
1
vote
2answers
35 views
$a \equiv b$ (mod $p$) implies $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?
Let $p$ be a prime number. If $a \equiv b$ (mod $p$), does that imply $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?
I think the answer will be yes, and I suspect that the way of proving it will involve ...
0
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2answers
73 views
Interpretation of Coefficients of Expanded Cyclotomic Polynomials
Working out the following definition of the Cyclotomic Polynomial
$$ {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),}
$$
you'll ...
1
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1answer
29 views
Unique subfield of order $2$ of $\mathbb{Q(\zeta_{7}})$ over $\mathbb{Q}$
Actualy this subfield is $\mathbb{Q}(i\sqrt{7})$ since $X^{2}+7$ is irreducible over $\mathbb{Q}$ and has $i\sqrt{7}$ as root.
My problem here is to show unicity, I tried something using the tower ...
0
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0answers
44 views
computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions
Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions.
If these have been computed before, I'm looking for a ...
1
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1answer
54 views
$p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Prove that $p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Here $\Phi_n(x)$ is nth cyclotomic polynomial.
I don't know what I should use. $$\Phi_n(x)=\prod_{{1\leq a\leq n } \& {(a,n)=1}}(x-\zeta_n^a) $$ or ...
3
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0answers
55 views
Question about Davenport's cyclotomy result.
In my Analytic Number Theory course, we are going through Davenport's Multiplicative Number Theory (3rd ed.), and I am having some trouble working through a certain part of section 3 (Cyclotomy). I ...
2
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1answer
57 views
Proving that $x^{2^n} + 1$ is irreducible in $Q[x]$
I've been working on this and this is my process:
I would like to use Eisenstein's criterion so I considered the substitution $y=x-1$. So
$$x^{2^n}+1=(y+1)^{2^n}+1=\sum_{k=0}^{2^n}{2^n \choose k}y^{2^...
2
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1answer
65 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
$$d = ...
3
votes
1answer
63 views
Why a certain integral domain is not a UFD.
Let
$$\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 ...
2
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1answer
42 views
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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0answers
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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 ...
3
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0answers
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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 ...
5
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0answers
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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 ...
0
votes
1answer
64 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\{ ...
2
votes
2answers
58 views
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, ...
1
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1answer
49 views
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 ...
3
votes
1answer
50 views
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
82 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{...
3
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1answer
49 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 ...
1
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1answer
90 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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2answers
446 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$,...
3
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1answer
59 views
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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0answers
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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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0answers
48 views
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\...
0
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1answer
72 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).$$
Since ...
4
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1answer
74 views
Do you know a generalization of a formula about cyclotomic polynomial?
Let n be a natural number. Then n-th cyclotomic polynomial is defined as follows:
$$\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
54 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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3answers
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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
42 views
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
42 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 ...
2
votes
1answer
49 views
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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1answer
31 views
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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0answers
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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 $\pmod p$...
0
votes
1answer
102 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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0answers
30 views
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 ...
1
vote
2answers
1k views
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{ ...
5
votes
1answer
64 views
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(...
1
vote
2answers
52 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 ...
0
votes
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:
...
1
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0answers
48 views
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):\...
0
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0answers
35 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 ...
