One Equals Minus One

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

Explanation 1
There are two values of $\sqrt {-1 \times -1}$.

We have:
 * $-1 = - \sqrt {-1 \times -1}$

We also have:
 * $1 = + \sqrt {-1 \times -1}$

The two are not the same thing.

Explanation 2
The property:
 * $\sqrt {a} \times \sqrt {b} = \sqrt {ab}$

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

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