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Find the cube root of this polynomial

Source: 2019 Jozsef Wildt International Math Competition-W. 13

May 18, 2020

determinantpolynomialcomplex numbersalgebra

Problem Statement

Let

a

b

and

c

be complex numbers such that

abc = 1

. Find the value of the cubic root of
\begin{tabular}{|ccc|}

b + n^3c

n(c - b)

n^2(b - c)

n^2(c - a)

c + n^3a

n(a - c)

n(b - a)

n^2(a - b)

a + n^3b

\end{tabular}