Error in proof: Distribution of exponents for negative number

Question

Here are steps of the "proof":

$1=1$

$\Rightarrow 1=\sqrt{1}$

$\Rightarrow 1=\sqrt{-1\times-1}$

$\Rightarrow 1=\sqrt{-1}\times\sqrt{-1}$

$\Rightarrow 1=i\times i$

$\Rightarrow 1=-1$

At which step did things go wrong?

Answer 1

It is no longer true that $$\sqrt{ab}=\sqrt{a}\sqrt{b}$$ when $a,b$ are not positive real numbers.

Additional remark: One must be careful when talking of $\sqrt{\cdot}$ in the realm of complex numbers, because it cannot be defined globally, i.e. there is no continuous function $$f:\Bbb C\to \Bbb C.$$ such that $f(z)^2=z$ for all $z\in\mathbb{C}$. Formally, we define $$\sqrt{z}=e^{\frac{1}{2}\log z}$$ where $\log$ is the complex logarithm defined on some proper simply connected open subset of $\Bbb C$.