So what is the point of being a field ?

See also Wedderburn's little theorem, which shows that any finite division ring is commutative and therefore a field. This is a pretty amazing result because rings were created partly to study algebra in a non-commutative setting, and many of the most important rings, such as n x n real matrices with n > 1, are non-commutative. The quaternions in particular are a non-commutative division algebra, not subject to the theorem because infinite.

The proof of Wedderburn's little theorem is relatively simple by the standards of professional math, but it's beyond me to even imagine ever coming up with it.

You can get that every integral domain is a field with fewer words by using a higher powered set theory result -- injections on finite sets are also surjections. The cancellation property says multiplication by any element is an injection, so it is also a surjection, i.e., 1 is in the range, so that gives you the multiplicative inverse.

> Let F be a field, and let a,b∈F such that ab=0. There are two cases to consider: a=0 and a≠0. If a=0, then indeed ab=0 by Proposition 1.

This part is a bit weird. If a=0, then we are already done, there's no need to prove ab=0 (which was already the assumption).

The other case can also be proved in a shorter way by just multiplying both sides of ab=0 with a^(-1) from the left.