For questions related to cyclotomic polynomials and their properties.

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1answer
66 views

How to solve the following solvable equation?

The following equation is solvable by radicals. How to get all the roots in terms of radicals? $ x^{12}+x^{11}+x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = 0$ The transcendental solution is given by ...
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0answers
21 views

Prove one of the cyclotomic polynomial identities

Let $\Phi_n(x)$ be the nth cyclotomic polynomial over $\mathbb Q$. $\Phi_n(x)=\frac{x^n-1}{\Pi_{d|n,d<n}\Phi_d(x)}$ for n>1, and $\Phi_1(x)=x-1$ Let $n=p_1^{r_1}...p_s^{r_s}$ with $p_i$ distinct ...
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1answer
34 views

$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
1
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1answer
25 views

Irreducibility of special cyclotomic polynomial.

I'm trying to show that $f(x)=1+x^p+x^{2p}+\dots +x^{p(p-1)}$ is irreducible over $\mathbb{Q}[X]$. I'm well aware that cyclotomic polynomials are irreducible, however the (many) proofs of this ...
5
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1answer
72 views

Units of $\mathbb Z[X]/(X^n+1)$?

What are the units of the cyclotomic ring $\mathbb Z[X]/(X^n+1)$, with $n$ being a power of $2$? I am starting to think that the set $\{\pm X^k,k=0,\dots,n-1\}$ contains all units, is that so ?
3
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1answer
29 views

Theorem on Repeating Decimals

So I am wondering if anyone recognizes the following theorem: Given a prime $p$, and a base $b$ (natural number $>1$), the period of $\frac{1}{p}$ expressed in base $b$ is the unique $d$ that ...
6
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1answer
114 views

Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of ...
3
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1answer
39 views

How to design a matrix with multiple chosen eigenvalues?

I want to "design" (build) a matrix that have multiple known eigenvalues. For these apriori chosen eigenvalues, I want to know the corresponding eigenvectors, too. The point is that I want to start ...
0
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1answer
42 views

Finding the $18$th cyclotomic polynomial $\phi_{18}(X))$.

I know that for an $n$th cyclotomic polynomial $\phi_n(X)$ the following equations hold: $x^n-1=\prod_{n_1|n} \phi_{n_1}(X)$ For $n=p$ prime, $\phi_p(X)=X^{p-1}+...+X+1$ So I used the following ...
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1answer
403 views

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then ...
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0answers
27 views

cyclotomic polynomial and unit group

Let $a$ ne a nonzero integer, p a prime, n a positive integer, and p does not divide n. Prove that $p | \Phi_{n}(a)$ if and only if a has period n in $(\mathbb{Z}/p\mathbb{Z})^{*}$ Actually I have ...
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0answers
171 views

solve in positive integers sum of squares of sines equation

Find all positive integer triples $(l,m,n)$ such that $\sin^2\frac{\pi}{n}+\sin^2\frac{\pi}{m}=\sin^2\frac{\pi}{l}$. I have found the solutions $(m,m,1)$ for any $m\in\mathbb{Z}^+$, and also ...
3
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1answer
66 views

Compute the minimal polynomial of $u$ over $\Bbb{Q}$ without using cyclotomic polynomials

Let $k\in \Bbb Z\setminus 7\Bbb Z$ and $a_k=\frac{2k\pi}{7}$. Compute the minimal polynomial of $u=$ over $\Bbb{Q}$ The "natural" (in my opinion) way to solve this problem, requires the use of ...
8
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2answers
150 views

Spotting that $\,x^8 + x^7 + 1\,$ is reducible.

I saw a puzzle recently that came down to spotting that $\,x^8 + x^7 + 1\,$ is reducible — namely, we have $$x^8 + x^7 + 1 = \left(1 + x + x^2\right)\left(1-x+x^3-x^4+x^6\right)$$ After playing ...
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0answers
30 views

Show that $f(x)=q(x)\Phi_p(x)^k$ implies $f(x^2)=r(x)\Phi_p(x)^k$ for some $q,r\in\mathbb{Q}[x]$ with gcd$(q,\Phi_p)=1$, $\gcd(r,\Phi_p)=1$.

Let $f\in\mathbb{Q}[x]$ and suppose that $f(x)=q(x)\Phi_p(x)^k$, for some $k\geq 1$ with $q\in\mathbb{Q}[x]$ and $\gcd(q(x),\Phi_p(x))=1$, where $\Phi_p$ is cyclotomic polynomial with $p>2$ ...
1
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1answer
49 views

roots of unity and cyclotomic polynomials over $\mathbb{F}_p$

Given a prime $p$, Let $n = p^d -1$ and let $f$ be an irreducible polynomial dividing the $n$-th cyclotomic polynomial in $\mathbb{F}_p[t]$. Let $\alpha = t + (f)$ in $\mathbb{F}_p[t]/(f)$ where $(f)$ ...
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0answers
30 views

Any formulæ for products of consecutive cyclotomic polynomials?

I am interested in any information about coefficients of the polynomials $\Psi_n(x):=\prod_{k=1}^n\Phi_k(x)$, where $\Phi_k$ are the cyclotomic polynomials. I realize of course that this is a ...
0
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1answer
64 views

Proving that cyclotomic polynomials have integer coefficients

I don't understand why Gauss's lemma is invoked in the proof in Dummit and Foote that $\Phi_n(x)$ (the $n$th cyclotomic polynomial) belongs to $\mathbb{Z}[x]$. I'm an analyst and I wanted to remind ...
2
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0answers
55 views

Lower bound for the values of cyclotomic polynomials evualuated at integers

Let $b,n \geq 2$ be integers and let $\Phi_n(b)$ be the value of the $n$-th cyclotomic polynomial evaluated at $b$. I've recently noticed by computer experiments that whenever $n$ is odd, we seem to ...
4
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1answer
45 views

Why is the sum of the ($\varphi(d)$-1)th coefficients of $\Phi_d$ equal to the ($\varphi$(n)-1)th coefficient of $\prod_{d\mid n} \Phi_d(X)$

I grasp this answer, except for one identity. To quote: "$\sum_{d\mid n}\left[\Phi_d(X)\right]_{-1} = \left[\prod_{d\mid n} \Phi_d(X)\right]_{-1}$" It isn't so simple i think because you don't take ...
4
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1answer
62 views

$\mathbb{Q}(\zeta)$ contains a unique subfield $K$ of degree $10$ over $\mathbb{Q}$?

Let $\zeta$ be a $151$th root of unity, $L = \mathbb{Q}(\zeta)$. How do I see that the cyclotomic field $L$ contains a unique subfield $K$ of degree $10$ over $\mathbb{Q}$? Can we conclude that ...
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2answers
96 views

Is $x^6-x^3+1$ irreducible over $\mathbb Q[x]$?

Is $x^6-x^3+1$ irreducible over $\mathbb Q[x]$? Approach If $x^6-x^3+1$ is reducible over $\mathbb Q[x]$, then it can be factored out with degree $1,2\;\text{or}\;3$. So check that $x^6-x^3+1$ has ...
1
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1answer
35 views

Splitting a polynomial into irreducible polynomials that all have the same degree

Let $q$ be a prime number and define $\Phi_q = X^{q-1} + \cdots + X^2 + X + 1 \in \mathbb{Z}[X] $. Let $p$ be a prime number and define $f_{q,p} = \Phi_q \bmod p \in \mathbb{F}_p[X]$. ...
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1answer
39 views

Where is this formula about cyclotomic polynomials proved?

In an article "Generalized Reciprocals, Factors of Dickson Polynomials and Generalized Cyclotomic Polynomials over Finite Field" by Fitzgerald and Yucas I see on page 18 in the proof of Lemma 7.2 (2) ...
1
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1answer
20 views

Erroneous calculation using self-reciprocity of cyclotomic polynomials?

Because of self-reciprocity of cyclotomic polynomials $\Phi_n(x)$ we have $$x^{\phi(n)}\Phi_n\left(\frac 1x\right)=\Phi_n(x)$$ with the Euler totient function $\phi(n)$. Now I concluded/calculated ...
0
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1answer
50 views

Irreducibility, Cyclotomic polynomial, How do Binomial Coefficients Simplify?

Let $p$ be prime and consider the polynomial $$f(x)=x^{p-1}+x^{p-2}+\dots+x^2+x+1 $$ Prove that $f(x)$ is irreducible Hint) May use without proof that $p|\binom {p} {a}$ with $a: 1\leq a \leq ...
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2answers
50 views

Showing 8th cyclotomic polynomial is irreducible

I'm reviewing my notes, and I'm not fully understanding an argument that the 8th cyclotomic polynomial is irreducible in $\mathbb{Q}[x]$. Here's the online of the argument ...
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0answers
34 views

cyclotomic polynomial $\Phi_{2n}(x)$ [duplicate]

Question is: "Show that $\Phi_{2n}(x)=\Phi_n(-x)$ for all odd numbers $n>1$" I try to prove this as follow, $\prod_{d \mid 2n} \Phi_d(x) =x^{2n}-1 = (x^n-1)(x^n+1) = -(x^n-1)((-x)^n-1) = ...
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2answers
57 views

What is the minimal polynomial of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$? [closed]

What is the minimal polynomial over $\mathbb{Q}$ of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$, where $\zeta_j$ is a $j$-th primitive root of unity for each $j$? I want to say it should be ...
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1answer
32 views

Example of splitting of prime ideals

I am trying to figure out how prime ideals of $\mathbb{Z}$ decompose in the Galois extension $\mathbb{Q}(\zeta_{7})/\mathbb{Q}$. And so, for this, I've picked the prime ideals $5\mathbb{Z}$, ...
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0answers
38 views

Show that $\frac{1-\zeta_p^k}{1-\zeta_p^j}$ is invertible in $\mathbb{Z}[\zeta_p]$

Let $p$ be a prime. For $1 \leq j,k \leq p-1$, show that $$\frac{1-\zeta_p^k}{1-\zeta_p^j}$$ is invertible in the ring $\mathbb{Z}[\zeta_p]$. First, I wanted to show that ...
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0answers
36 views

Generator for a cyclotomic subextension

Consider the Galois cyclotomic extension $\mathbb{Q}(\zeta_{7})/\mathbb{Q}$. I've found via Galois theorem the following intermediate field of degree 2 \begin{equation} ...
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2answers
87 views

Splitting of primes in an extension with “unknown” ring of integers

Consider the Galois extension $\mathbb{Q}(\zeta_{7})/\mathbb{Q}$. I am looking for the decomposition of the prime ideal $5\mathbb{Z}$ in the integral closure of $\mathbb{Z}$ in ...
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0answers
69 views

How to think about Galois groups in a geometric way?

First of all, I've read some questions that have similar title, but I didn't find any answers that correspond to what I have in mind. So, I am an undergraduate student starting my investigations in ...
3
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1answer
92 views

Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$. Attempt: I am confident that I computed $|E|$ correctly, but I am ...
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0answers
24 views

Is the longest chain of non-increasing values in this sequence related to cyclotomic polynomials unbounded?

The $n$th order cyclotomic polynomial is defined as $$ \Phi_n(x) = \prod_{\substack{1\le k\le n \\ \gcd(k, n) = 1}}{\left(x - e^{2i\pi k/n}\right)} $$ Define $c_n$ to be the smallest integer $m$ such ...
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3answers
48 views

$\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$

Show that $\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$, where $z_0,...,z_{n-1}$ are the $n$-th roots of unity. For $k=n$ it ...
3
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1answer
58 views

The largest subset of a finite cartesian product in which distinct elements differ in at least 2 components

Let $A_1,\ldots,A_n$ be finite sets of sizes $a_1,\ldots,a_n$. What is the largest possible size of a subset $S\subset\bigotimes A_k$ such that if $(d_1,\ldots,d_n),(e_1,\ldots,e_n)\in S$, then ...
18
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1answer
265 views

What is the number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
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4answers
60 views

determining which cyclotomic polynomial is $x^8 -x^4+1$

Given that the following polynomial\begin{equation*}f(x)=x^8-x^4+1\end{equation*} is a cyclotomic polynomial $\Phi_n$ for some $n\in \mathbb N$. are there some basic tools to determine $n$? I know ...
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1answer
59 views

$p$-adic Euler's totient function

Let $p$ be a prime number, $\overline{\mathbb{Q}_p}$ a fixed algebraic closure of $\mathbb{Q}_p$. Let $\alpha$ be a primitive $n$-th root of unity in $\overline{\mathbb{Q}_p}$ and $d$ its degree over ...
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0answers
65 views

A question about the degree of sin(2π/n) over the rationals

In cyclotomic theory, $\cos (2\pi /n)$) is shown to have degree $\varphi (n)/2$ over the rationals $\mathbb{Q}$, while $\sin (2\pi /n)$has degree $\varphi (n)$ (as long as n is not divisible by ...
8
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2answers
99 views

Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
1
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1answer
71 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
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1answer
51 views

Prove that $Φ_{nm}(x) = Φ_n(x^m)$ if every prime divisor of m is also divisor of n

Let $m$ and $n$ be natural numbers that every prime divisor of $m$ is also a divisor of $n$. We can define $Φ_{ab}(x)$ for every prime $a>0$ like this: $$Φ_{ab}(x) = \begin{cases} Φ_b(x^a), & ...
0
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1answer
33 views

Find a polynomial in $\mathbb{Z}_{41}$

Find a $7^{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}$, so that $$ p(14^i) = i\ (mod\ 41)\ \forall i = 0,1,\ldots,7. $$ $3$ is the $8^{th}$ primitive root of unity and $3 * 14 = 8 * 36 = 1$ ...
1
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1answer
35 views

Is there a relationship between these polynomial concepts?

I'm currently doing a bit of reading on abstract algebra (more specifically Polynomial Theory), and noticed something that may have some sort of significance perhaps? The section I'm reading on at ...
3
votes
2answers
202 views

A unit of seventh cyclotomic field

I have troubles with the following problem about units. Show that $1+\zeta $, $1+\zeta+\zeta^2$ are units in the field $\mathbb{Q[\zeta]}$, where $\zeta$ is a seventh primitive root of unit ...
0
votes
0answers
17 views

Do the minimal cyclotomic polynomials always have coefficient 1, 0 or -1?

I noticed the following trend in the minimal cyclotomic polynomials for each $\zeta_{k}$. $\zeta_{5}\rightarrow t^4+t^3+t^2+1\\\zeta_{6}\rightarrow t^2-t+1\\\zeta_{7}\rightarrow ...
2
votes
1answer
52 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...