[carloswilson]

> 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?

[Koshkin]

To be fair, you can have a field that only has integers. For example, Z mod 5 is a field.

[carloswilson]

I am aware. It is typically represented as Z_5. They are called prime fields.

I highly doubt wsxcde meant prime fields in their comment though. wsxcde seemed to be talking about the set of all integers and the set of all real numbers in their comment. Only the latter is a field (and a ring) whereas the former is only a ring.

And (-a)(-b) = ab holds in rings (and thus fields).