Effect of squaring while finding roots of unity

Question

Consider $$b=\frac{1}{b}\rightarrow b^2=1$$ Clearly $b=\pm1$   But if we square the above equation on both sides and then solve $$(b=\frac{1}{b})^2\rightarrow b^2=\frac{1}{b^2}\rightarrow b^4=1 $$ And we know fourth root of unity are $1,i,-1,-i$     why am i getting extraneous roots can someone please explain

Answer 1

Because $f(x)=g(x)$ implies $[f(x)]^2=[g(x)]^2$, but not vice versa. More generally, $y^2=z^2$ doesn't imply $y=z$. That's why you wrote $b=\color{blue}{\pm}1$ in the first place.