What's wrong with this proof of 1 = -1?

$\displaystyle -1$	$\displaystyle = -1$
$\displaystyle \frac{-1}{1}$	$\displaystyle = \frac{1}{-1}$
$\displaystyle \sqrt{\frac{-1}{1}}$	$\displaystyle = \sqrt{\frac{1}{-1}}$	Take the square root of both sides.
$\displaystyle \frac{\sqrt{-1}}{\sqrt{1}}$	$\displaystyle = \frac{\sqrt{1}}{\sqrt{-1}}$	Since $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ .
$\displaystyle \frac{i}{1}$	$\displaystyle = \frac{1}{i}$	Since $i=\sqrt{-1}$ .
$\displaystyle i^2$	$\displaystyle = 1$	Cross-multiply
$\displaystyle -1$	$\displaystyle = 1$	Since .

The erroneous step was assuming $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ always holds when a or b is negative. An argument can be made that while taking the square root of both sides, the ``negative'' root was not considered, else we would have concluded

to be valid and

to be invalid, i.e. extraneous. The aforementioned erroneous assumption is the reason

is extraneous.