I've a homework question, stating: Let $F$ be a field with the addition and multiplication operation such that $\mathrm{char}(F)=2$. For $a,b\in F$, prove that $a + b = a-b$.
I know that $\mathrm{char}(F)=2$ means that for $F, 1+1 = 0$, but I can't see how that helps me to solve the problem.
Can I assume that $a,b$ must be $0$ or $1$? I think that's wrong. What else can I deduce that'll help me?
$$a+b+b=a+b(1+1)=a+b\cdot 0=a$$ Therefore $$a+b=a-b$$