(a) 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 .

Prove that $\mathbb{F}^{2}$ can be made into a field, with operations

$$(x, y)+(z, w)=(x+z, y+w)$$

and

$$(x, y) \cdot(z, w)=(x z-y w, x w+y z) .$$

(b) Let $p$ be a prime of the form $4 m+3$. Prove that $-1$ is not a square $(\bmod p)$, and deduce that there exists a field with exactly $p^{2}$ elements.