> The set over which the field is defined (Z or R)
What? The set of all integers, Z, is not a field! R is. But Z isn't. Z is a ring, a commutative ring. I doubt you understand what a field is!
> already contains -3, -2 etc.
Yes, and those elements are literally the additive inverses of their positive counterparts. If you disagree with this, then the numbers -3, -2, etc. literally have no meaning.
> It then turns out that -3 is the additive inverse of 3.
Are you making this all up with your original research or do you have any proper literature written by a professional mathematician to back it up?
What? The set of all integers, Z, is not a field! R is. But Z isn't. Z is a ring, a commutative ring. I doubt you understand what a field is!
> already contains -3, -2 etc.
Yes, and those elements are literally the additive inverses of their positive counterparts. If you disagree with this, then the numbers -3, -2, etc. literally have no meaning.
> It then turns out that -3 is the additive inverse of 3.
Are you making this all up with your original research or do you have any proper literature written by a professional mathematician to back it up?