Because roots of unity is too mainstream

Algebra Level 5

Consider the set of all complex $n^\text{th}$ roots of $x$ , $\displaystyle \{a_i\}_{i=1}^{i=n}$ where $x\in\Bbb{C}$ .

We further denote the summation $\displaystyle P_m = \sum_{j=1}^n \left(a_j\right)^m$ .

Find an expression for $P_{nk}$ .

Note: $m,n,k$ are all positive integers.

Bonus: Can you write a result which provides all values of $P_m$ ?

Clarification: $\Bbb C$ is the set of all complex values.

Inspired by this problem.

x^{nk}

x

n

x^k

nx^k

x^n

knx

k^n

2 solutions

Tijmen Veltman
May 24, 2015

We know that $a_i^n=x$ for all $i$ , therefore:

$P_{nk}=\sum_{j=1}^n (a_j)^{nk} =\sum_{j=1}^n (a_j^n)^k =\sum_{j=1}^n x^k =\boxed{nx^k}.$

Nicely done. Now, try the bonus problem. :D

Prasun Biswas - 6 years ago

Wilmar Bolanov
May 29, 2015

For all $m \neq kn$ $P m = 0$ and if $m = kn$ then $P m = nx^k.$

Yes, that is indeed the answer to the bonus question but don't you think you should post the proof of that result too?

Otherwise, this solution is incomplete.

Prasun Biswas - 6 years ago