What is a complex root?

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Answer 1 · 2015-11-01T09:26:11

${z in \mathbb{C} ; f(z) = 0}$

$f(x) = x^3 - 1$

The roots of $f$ under a domain $A$ are ${x in A ; f(x) = 0}$ .

What are the real roots of $f$ ? ${x in \mathbb{R} ; x^3 = 1} = {1}$ .

But there are two other complex roots:

$\frac{x^3 - 1}{x - 1} = x^2 + x + 1 = 0$

$x_± = \frac{-1 ± i sqrt {3}}{2}$

Therefore, three are complex roots of $f$ .

${x in \mathbb{C} ; x^3 = 1} = {1, x_+, x_-}$ .