Each $x \in \operatorname{Spec} A$ is a prime ideal, hence $A / x$ has only two elements $0$ and $1$. Now define $\text{Frac}(A / x)$ can see only $1$ can be put at denominator position, hence we get a ring with only elements $\{ 0/1, 1/1 \}$, which must be $\mathbb{F}_2$ again.