For questions related to cyclotomic polynomials and their properties.

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1answer
39 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
48 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
40 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
46 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 ...
1
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2answers
83 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
33 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 $f_{q,p} = \Phi_q \bmod p \in \mathbb{F}_p[X]$. ...
1
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1answer
32 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
19 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 ...
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0answers
22 views

Show $1+\zeta_7+\zeta_7^6$ and $1+\zeta_7^3+\zeta_7^4$ are multiplicatively independent

I need to show that $1+\zeta_7+\zeta_7^6$ and $1+\zeta_7^3+\zeta_7^4$ are multiplicatively independent, that is: $$\nexists \; a,b \gt 0: (1+\zeta_7+\zeta_7^6)^a*(1+\zeta_7^3+\zeta_7^4)^b=1$$ I ...
0
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1answer
38 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
43 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 ...
0
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0answers
30 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) = ...
1
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2answers
54 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 ...
2
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1answer
27 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 $5\mathbb{Z}$, ...
1
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0answers
35 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 ...
2
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0answers
34 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} ...
2
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2answers
78 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 ...
3
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0answers
60 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
76 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
23 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 ...
2
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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
56 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
241 views

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 ...
0
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4answers
58 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 ...
1
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1answer
49 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 ...
0
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0answers
59 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
90 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 ...
0
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1answer
46 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
32 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
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2answers
186 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 ...
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0answers
15 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
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1answer
48 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 ...
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2answers
62 views

For any integer $n$ there exists a prime $p$ such that the group $Z_p^*$ contains an element of order $n$

Prove that for any integer $n$ there exists a prime $p$ such that the group $\mathbb Z_p^*$ contains an element of order $n$. Show that this is possible only if $p \equiv 1\pmod{n}$. And how can I ...
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1answer
23 views

Primitive roots of unity proof verification

"Let $C_n(x)$ be the polynomial such that the roots of $C_n(x)=0$ are the primitive $n^{th}$ roots of unity. Prove that there are no positive integers $q,r,s$ for which $C_q(x)=C_r(x)C_s(x)$." My ...
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3answers
65 views

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$ Note: $\Phi_n(x)$ is the $n$th cyclotomic polynomial whose roots are the primitive $n$th roots of unity if n is odd then $-1$ cannot ...
3
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0answers
99 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
1
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1answer
39 views

Irreducible polynomials in $\Bbb F_{2}[x]$ and their generators.

Hi am trying to find as many irreducible polynomials in $\Bbb F_{2}[x]$ as possible and their generating cyclotomic cosets. So far I have found 3 - a number am quite pleased with; $1 + x + x^2 + x^3 + ...
2
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2answers
61 views

Let $\zeta=e^{2\pi i/n}$ Prove that $x^n -1 =(x-1)(x-\zeta)(x-\zeta^2) \dots (x-\zeta^{n-1})$

This is a question about cyclotomic polynomials and I have already shown that $x^n-1 =\Pi\Phi_d(x)$, taking the product over all divisors d of n.
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3answers
53 views

Roots of “almost cyclotomic” polynomials

While looking at the behavior of the probability of getting a run of $k$ heads during $n$ tosses of a fair coin, I ended up needing to know the nature of the roots of polynomials of the form $$ P(x) = ...
0
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1answer
45 views

Q basis for splitting field

I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it ...
0
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0answers
33 views

Solution to $(x+1)^M-2=0\mod(x^a-1,M)$ with $a|M$.

My exercise sheet requires me to look at the following. I am looking for an elementary solution to (all polynomials have integer coefficients): Do there exist cases of a>1 and M integers with a ...
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1answer
54 views

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$.

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$. How to derive this. I am having no clue. I guess $n|m$. But what is the proof?
0
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1answer
58 views

Let $G$ be a finite abelian group. Show that there is a Galois extension $K/\Bbb Q$ with $\text{Gal}(K/\Bbb Q) \cong G$. [duplicate]

Let $G$ be a finite abelian group. Show that there is a Galois extension $K/\Bbb Q$ with $\text{Gal}(K/\Bbb Q) \cong G$. I have seen one proof using For a fixed positive integer $n$, there are ...
0
votes
0answers
28 views

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$ [duplicate]

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$. I have thaught that for $\Bbb Q(\sqrt 3)$ is in $\Bbb Q(w)$ where $w$ is in the $3$rd root of unity. ...
1
vote
0answers
60 views

Cyclotomic polynomial, after adjoining a radical

Suppose that $p>2$ is prime and that $a$ is a rational number for which $\sqrt[p]{a}$ is in $\mathbb C\backslash\mathbb Q$. The cyclotomic polynomial $\Phi_p$ is well-known to be irreducible over ...
1
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0answers
53 views

Lucas's Cyclotomic Formula

There are 2 well-known formulas involving Cyclotomic polynomials, which can be described roughly as writing $\phi_n$ as norms of elements in some quadratic extension. They appear in wikipedia and ...
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2answers
36 views

Why does the automorphism mapping $\omega$ to 1, not an element of Galois group

Let $L/\mathbb{Q}$ be a field extension, where $L=\mathbb{Q}(\omega)$ and $\omega=e^{\frac{2\pi i}{7}}$. In my textbook it states that $Aut(L/\mathbb{Q})=\{\sigma_i|\sigma_i(\omega)=\omega^i, 1\leq ...
1
vote
1answer
43 views

Irreducibility of cyclotomic polynomials of prime order

I am stuck with an exercise, where I have to prove the $\Phi_5[X] \in \mathbb{F}_2(x)$ is irreducible. I know that $X^5-1=\Phi_5(X)(X-1)$ (shown in previous part of the exercise) $X^2+X+1$ is the ...