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.


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