Show that 0 cannot have a multiplicative inverse in any of our number systems.
We will need the fact that $\displaystyle 0\cdot x=0$ for all x.
This can be proved if needed.
Now supose that $\displaystyle 0$ does have an inverse. Call it $\displaystyle e$
Then by defintion of multiplicive inverse
$\displaystyle e \cdot 0= 0 \cdot e=1$
But by our theorem above $\displaystyle 0=e\cdot 0 =1$ this is a contradiction. So the assumption that zero has an inverse must be false.