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Tagged with primitive-roots cyclotomic-polynomials
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Cyclotomic Polynomials and The Existence of Infinite Prime Power
Prove that there exist infinitely many positive integers n such that all prime divisors
of $n^2 + n + 1$ are not greater than $\sqrt{n}$
This is a problem related to cyclotomic polynomial. It is ...
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Prove that the roots of cyclotomic polynomial $\Phi_{p-1}(x) \equiv 0 (mod~p)$ are exactly the primitive roots mod p
$p$ is a prime, and $\Phi_{p-1}(x)$ denote the cyclotomic polynomial of order $p-1$. And I want to show the following:
$g$ is a solution of the congruence $\Phi_{p-1}(x) \equiv 0 (mod~p)$ if and only ...
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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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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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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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On multiplicative and additive properties of cyclotomic polynomials
Is there explicit relation between $\Phi_{a+b}(x)$, $\Phi_{ab}(x)$, $\Phi_{a}(x)$ and $\Phi_{b}(x)$ at general coprime or non-coprime $a,b\in\Bbb Z$?
If $a,b$ are distinct primes then we have $x^{ab}-...
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Why is it the case that if $a$ is a primitive root of $x^p=1$, then $\frac{x^p-1}{x-1}=(x-a^2)(x-a^4)...(x-a^{2(p-1)})=1+x+x^2+...+x^{p-1}$?
If $p$ is an odd prime. Why is it the case that if $a$ is a primitive root of $x^p=1$, then $\frac{x^p-1}{x-1}=(x-a^2)(x-a^4)...(x-a^{2(p-1)})=1+x+x^2+...+x^{p-1}$? I can see why $\frac{x^p-1}{x-1}=1+...
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Eigenvalues are roots of cyclotomic polynomial
I am reading Lyndon and Shupp's 'combinatorial group theory'. At page 25 it is stated that
if $g$ is an element of finite order $n$ in $\mathbb{GL}(2, \mathbb{Z})$, its eigenvalues must be roots ...
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Finding the splitting field of $\Phi_{21}(x)$ over $\mathbb Q$
In another question
I asked how I would find the miminal polynomial of a primitive nth root of unity over $\mathbb Q$, which was very well answered and easy to follow.
Taking the same example, let $...
user484410
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Evaluate $\prod_{k=1}^{n-1} \cos \frac{k \pi}{n}$ where $\text{gcd}(n,k)=1$ and $n$ is odd
Prove that $\prod\limits_{1 \le k \le n-1,\gcd(n,k)=1} \cos \frac{k \pi}{n}=\frac{(-1)^{\varphi(n)}}{2^{\varphi (n)}}$ where $n$ is an odd number.
I used the same method as here but how can we ...
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How to show the degree $[\Bbb Q(\alpha _n+\frac{1}{\alpha_n}) : \Bbb Q]$ is $\phi(n)/2$, where $\alpha_n$ is a primitive $n$-th root of unity?
How to show the degree $[\Bbb Q(\alpha _n+\frac{1}{\alpha_n}) : \Bbb Q]$ is $\phi(n)/2$, where $\alpha_n$ is a primitive $n$-th root of unity?
I already know that $[\Bbb Q(\alpha _n ):\Bbb Q]$ is ...
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Irreducibility of $X^5-7$ over $\mathbb{Q}(\sqrt[7]{2})[X]$ and degree of spitting field
I have worked through these two questions but am unsure if I got the right idea, please may you help me?
Prove that $X^5-7$ is irreducible over $\mathbb{Q}(\sqrt[7]{2})[X]$
Can we say that $f(X)=X^5-...
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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 ...
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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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Find a polynomial in $\mathbb{Z}_{41}$
Find a $7^\text{th}$ degree polynomial $p(x)$ in $\mathbb{Z}_{41}[x]$, so that
$$
p(14^i) = i\pmod{41}\ \forall i = 0,1,\ldots,7.
$$
Hint: $3$ is the $8^\text{th}$ primitive root of unity and $3 \...
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