lean_workbook_plus_75553

Let $\mathbb{F}$ be a field such that the equation $x^2 = -1$ has no solution in $\mathbb{F}$ . Prove that if $x$ and $y$ are elements of $\mathbb{F}$ such that $x^2 + y^2 = 0$ , then both $x$ and $y$ must equal $0$ .