Roots of Complex Number/Examples/z^4 - 81 = 0

Theorem
The roots of the polynomial:
 * $z^4 - 81$

are:
 * $\set {3, 3 i, -3, -3 i}$

Proof
From Factorisation of $z^n - a$:
 * $z^4 - a = \ds \prod_{k \mathop = 0}^3 \paren {z - \alpha^k b}$

where:
 * $\alpha$ is a primitive complex $4$th root of unity
 * $b$ is any complex number such that $b^4 = a$.

Here we can take $b = 3$, as $81 = 3^4$.

Thus:
 * $z = \set {3 \exp \dfrac {k i \pi} 2}$