Solve $x \times x = 1$ in a field?

Question

If I am in an arbitrary field and I have $x \times x = 1$, does this imply $x = 1$ or $x = -1$?

I feel like it should and I have been trying to prove it but I can't get there. Or maybe there is an obscure field where this is not true.

Thanks

Answer 1

In fact, the equation $x^2=1$ has at most $2$ solutions in a field, so the only solutions are $x=1$ and $x=-1$. If the characteristic of the field is $2$, we only have one solution namely $x=1$

Answer 2

In any field (even a non commutative field) $(x+1)(x+(-1))=x^2-x+x-1=x^2-1$, therefore $$x^2-1\iff x=-1\text{ or }x=1$$ If the characteristc of the field is $2$ both solutions are the same.

This reasoning also works in any integral domain.