field are domains.

Question

I want to prove that field are domain. Let $K$ a field. Suppose it's not a domain. Then there are $a,b\in K\setminus\{0\}$ s.t. $ab=0$. Let $a'$ and $b'$ s.t. $a'a=aa'=1$ and $b'b=bb'=1$. Then $$1=(a'a)(bb')=a'(ab)b'=0,$$ which is impossible. Does it work ?

Answer 1

Yes, it's fine.

A shorter proof would be: if $a'$ is such that $a'a=1$, then$$0=a'(ab)=(a'a)b=b.$$