Proof that 1=-1 using algebra

Proof that -1 = 1

Step 1:  -1 / 1 = 1 / -1     Looks harmless
Step 2:  ( -1 / 1 ) ^{1 / 2} = ( 1 / -1 ) ^{1 / 2}     After taking the square root
Step 3:  ( -1 ) ^{1 / 2} / ( 1 ) ^{1 / 2} = ( 1 ) ^{1 / 2} / ( -1 ) ^{1 / 2}     After splitting terms
Step 4:  i / 1 = 1 / i     After simplifying, using (-1) ^{1 / 2} = i
Step 5:  i = 1 / i
Step 6:  -1 = 1           On multiplying by i

Explanation

Step 4 is wrong because ( -1 ) ^{1 / 2} is i or -i. Note that i ² = ( - i ) ² = - 1.
Similarly, ( 1 ) ^{1 / 2} is 1 or -1.
So step 4 should read (+ or -) i /( (+ or -) 1 ) = (+ or -) 1 / ( (+ or -) i ), which resolves to:
(+ or -) 1 = (+ or -) 1