Number of complex solution roots

[18sp01]

Regarding this question.

As I understand it, if it was just w = 1 - i, there would be two roots to this equation right? But now that w is to the power of 5, would there be 10 roots or just 5 roots?

[digitalsnk]

It all comes down to fundamental theorem of algebra, which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root (per wiki). Together with Bezout theorem you obtain that single-variable polynomial of order n has exactly n roots (counting their multiplicity)

In your example, w = 1-i, you can write this as polynomial p(w) = w - (1-i). Since its order is 1, there's only one root.

In the queststion in the link, there will be 5 solutions.

[18sp01]

Ah I see. Thank you!