Questions tagged [cyclotomic-polynomials]
For questions related to cyclotomic polynomials and their properties.
0
votes
1answer
32 views
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 $E=\mathbb{Q}(\zeta+\zeta^{2}+\zeta^{4})$ ...
2
votes
1answer
49 views
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 ...
1
vote
2answers
45 views
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_{(...
1
vote
1answer
43 views
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 $x^p - 1 = (x-1)^p$.
...
0
votes
0answers
25 views
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 ...
1
vote
2answers
25 views
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 ...
0
votes
1answer
44 views
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^{...
0
votes
0answers
24 views
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 ...
1
vote
0answers
29 views
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}}\...
1
vote
0answers
31 views
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 ...
3
votes
0answers
35 views
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 $x^4+1$ (...
2
votes
0answers
74 views
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 ...
2
votes
1answer
65 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 $...
0
votes
1answer
36 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
vote
0answers
20 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:
$$\begin{vmatrix}
0&1&0&\dots& 0 \\
0&0&1&\dots&...
0
votes
1answer
37 views
Divisors of a cyclotomic polynomial
I am not entirely sure of the validity of this source (https://proofwiki.org/wiki/Prime_Divisors_of_Cyclotomic_Polynomials), but it claims that all prime divisors $d$ of $\Phi_n(a)$ have $d\equiv 1\...
0
votes
0answers
40 views
Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$
i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$.
...
5
votes
1answer
103 views
minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$
How to find the minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$ ? And the minimal polynomial of $\sin (2\pi/11)$ over $\mathbb Q(\sqrt {11})$ ?
I know that the minimal polynomial of $...
1
vote
1answer
46 views
$g^{0p^nk}+g^{p^nk}+g^{2p^nk}+…+g^{(p-2)p^nk} \equiv 0$ (mod $p^n)$ if $p-1$ doesn't divide $k$
Let $p$ be prime and $n\geq 1$. Let $g$ be the integer equivalent of some generator for $(\mathbb{Z}/p\mathbb{Z})^ \times$.
Let $k\in \{0,1,2,...\} \subseteq \mathbb{Z}$ such that $p-1$ does NOT ...
1
vote
2answers
44 views
$a \equiv b$ (mod $p$) implies $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?
Let $p$ be a prime number. If $a \equiv b$ (mod $p$), does that imply $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?
I think the answer will be yes, and I suspect that the way of proving it will involve ...
0
votes
2answers
81 views
Interpretation of Coefficients of Expanded Cyclotomic Polynomials
Working out the following definition of the Cyclotomic Polynomial
$$ {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),}
$$
you'll ...
1
vote
1answer
33 views
Unique subfield of order $2$ of $\mathbb{Q(\zeta_{7}})$ over $\mathbb{Q}$
Actualy this subfield is $\mathbb{Q}(i\sqrt{7})$ since $X^{2}+7$ is irreducible over $\mathbb{Q}$ and has $i\sqrt{7}$ as root.
My problem here is to show unicity, I tried something using the tower ...
0
votes
0answers
44 views
computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions
Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions.
If these have been computed before, I'm looking for a ...
1
vote
1answer
54 views
$p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Prove that $p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Here $\Phi_n(x)$ is nth cyclotomic polynomial.
I don't know what I should use. $$\Phi_n(x)=\prod_{{1\leq a\leq n } \& {(a,n)=1}}(x-\zeta_n^a) $$ or ...
3
votes
0answers
61 views
Question about Davenport's cyclotomy result.
In my Analytic Number Theory course, we are going through Davenport's Multiplicative Number Theory (3rd ed.), and I am having some trouble working through a certain part of section 3 (Cyclotomy). I ...
2
votes
1answer
79 views
Proving that $x^{2^n} + 1$ is irreducible in $Q[x]$
I've been working on this and this is my process:
I would like to use Eisenstein's criterion so I considered the substitution $y=x-1$. So
$$x^{2^n}+1=(y+1)^{2^n}+1=\sum_{k=0}^{2^n}{2^n \choose k}y^{2^...
2
votes
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
$$d = ...
3
votes
1answer
89 views
Why a certain integral domain is not a UFD.
Let
$$\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
votes
1answer
48 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 ...
0
votes
0answers
44 views
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
votes
0answers
58 views
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
votes
0answers
67 views
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
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\{ ...
2
votes
2answers
67 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
vote
1answer
57 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
53 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+\...
0
votes
1answer
153 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
votes
1answer
51 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
vote
1answer
110 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 ...
13
votes
2answers
475 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
votes
1answer
81 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$ ...
2
votes
0answers
25 views
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 ...
0
votes
0answers
61 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
votes
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).$$
Since ...
4
votes
1answer
76 views
Do you know a generalization of a formula about cyclotomic polynomial?
Let n be a natural number. Then n-th cyclotomic polynomial is defined as follows:
$$\Phi_{n}(x)=\prod_{k\in\mathbb{N}_{<k},(n,k)=1}(x-\zeta^k)$$
where $\mathbb{N}_{<k}$ means the set of natural ...
1
vote
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 ...
0
votes
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 $...
3
votes
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 ...
0
votes
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 ...
0
votes
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 ...
