A question in Rotman's Advanded Modern Algebra asks to prove the question in the title. I'm convinced of my proof, but a subquestion asked to prove that $1+1$ is zero, and for this I proceeded on a case by case basis; that is, assuming for the sake of contradiction that $1+1 \ne 0$, given that we now know that the field $F = \{0,1,1+1,a\}$, I proved that this structure cannot be a field. 

However, I think this solution is quite ugly and I was wondering if is a nicer but still elementary solution (avoiding the fact that char($F$) = $2$).

PS I'm not sure if this is a suitable question. If it's not, feel free to remove it.