Something slightly bothering me: for a field $k$, with closure $\overline{k}$, polynomial rings $k[x,y]$ and $\overline{k}[x,y]$, say we have $\overline{M}$ a maximal ideal of $\overline{k}[x,y]$ and $M := \overline{M} \cap k[x,y]$.
Why does $[\overline{M} \neq \overline{k}[x,y]] \Rightarrow [M \neq k[x,y]]$ hold?