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n-th roots of unity (some equivalences)

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1

$begingroup$

Let $q$ be a prime power and $n$ a positive integer s.t. $gcd(n, q) = 1$.
Let $E(n)$ be the set of the complex $n$-th roots of unity. For every positive integer $d$ such that $d | n$ , let

$$Q_d(x) := prod_alpha in E(n):mathrmord(alpha)=d (x- alpha)$$

Now I've got a (maybe simple and easy...) question: if $p$ is prime and $m$ a positive integer coprime to $p$. In that case, why is it true that $Q_p(x) = 1 + x + x^2 + ... + x^p-1$ ?

Thanks for any hint!

linear-algebra abstract-algebra

share|cite|improve this question

asked Mar 29 at 11:37

JohnDJohnD

341112

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Cyclotomic polynomial?
  $endgroup$
  – Thomas Shelby
  Mar 29 at 11:42
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $q$ be a prime power and $n$ a positive integer s.t. $gcd(n, q) = 1$.
Let $E(n)$ be the set of the complex $n$-th roots of unity. For every positive integer $d$ such that $d | n$ , let

$$Q_d(x) := prod_alpha in E(n):mathrmord(alpha)=d (x- alpha)$$

Now I've got a (maybe simple and easy...) question: if $p$ is prime and $m$ a positive integer coprime to $p$. In that case, why is it true that $Q_p(x) = 1 + x + x^2 + ... + x^p-1$ ?

Thanks for any hint!

linear-algebra abstract-algebra

share|cite|improve this question

asked Mar 29 at 11:37

JohnDJohnD

341112

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Cyclotomic polynomial?
  $endgroup$
  – Thomas Shelby
  Mar 29 at 11:42
  

  
  
  
  

add a comment | 

$begingroup$

Let $q$ be a prime power and $n$ a positive integer s.t. $gcd(n, q) = 1$.
Let $E(n)$ be the set of the complex $n$-th roots of unity. For every positive integer $d$ such that $d | n$ , let

$$Q_d(x) := prod_alpha in E(n):mathrmord(alpha)=d (x- alpha)$$

Now I've got a (maybe simple and easy...) question: if $p$ is prime and $m$ a positive integer coprime to $p$. In that case, why is it true that $Q_p(x) = 1 + x + x^2 + ... + x^p-1$ ?

Thanks for any hint!

linear-algebra abstract-algebra

share|cite|improve this question

asked Mar 29 at 11:37

JohnDJohnD

341112

$endgroup$

share|cite|improve this question

asked Mar 29 at 11:37

JohnDJohnD

341112

share|cite|improve this question

asked Mar 29 at 11:37

JohnDJohnD

341112

asked Mar 29 at 11:37

JohnDJohnD

341112

asked Mar 29 at 11:37

JohnDJohnD

341112