One Equals Minus One

Resolution
This is a falsidical paradox arising from incorrect reasoning about the nature of square roots.

Explanation
The property:
 * $\sqrt a \times \sqrt b = \sqrt {a b}$

can only be used when:
 * $a \ge 0$ and $b \ge 0$

It is specifically false when both $a$ and $b$ are negative.

Since $-1 < 0$, this property of square roots cannot be applied to this statement.

This is how the analysis is to go:

and all is well with the world.

Also see

 * Definition:Complex Number