[JadeNB]

> It's not a convention that cancellation holds for fields. It's a theorem one proves about integral domains. That a field is closed under multiplication is not a theorem; it's part of the definition of being a ring.

Yes, that's what I said. My point was that, between

(A) making an axiom the right to divide in a field by non-0 numbers, and proving the closure of non-0 numbers in a field under multiplication,

and

(B) just making an axiom the closure of non-0 numbers in a field under multiplication,

it is only convention (rooted in the deeper convention of not making an additional axiom out of something we could prove from existing ones) that we choose (A) instead of (B). (Also notice that I was talking about the closure under multiplication of the set of non-0 numbers, which is (usually) not part of the definition, rather than of the entire set, which is indeed always part of the definition.)