Let $k$ be any field. Fix an algebraic closure $\overline{k}$ and let $k_s$ denote the separable closure of $k$ within $\overline{k}$. Define an étale $k$-scheme as one with discrete underlying space such that for every $x\in X$, the local ring $\mathcal{O}_{X,x}$ is a field, which is a finite separable extension of $k$.

- Is $X$ affine?
- $X$ is a disjoint union of spectrums of finite separable extensions of $k$;
- $X(k_s) \cong X(\overline{k})$.

I think that 1. follows from 2. for if $X=\sqcup_i Spec(K_i)$ with $K_i$ a finite separable extension of $k$, then $X=Spec (\prod_i K_i)$ (by extending Exercise II.2.19 in Hartshorne to the possibly infinite case) and $X$ is affine. For 3. I only know that $k_s\subseteq \overline{k}$ implies that the canonical map $X(k_s)\to X(\overline{k})$ is injective since $X$ is separated. However I'm having no luck showing the other "inclusion."

Thank you for your help.