The definition is that it is the additive identity for the field; eg x + a = x no matter what value x takes and what field you are considering. This must be uni

The definition is that it is the additive identity for the field; eg x + a = x no matter what value x takes and what field you are considering. This must be unique; suppose a and b are both additive identities for a field, then b + a = b and a + b = a, but commutativity gives us a + b = b + a, resulting in a = b. The reason the additive identity cannot have a multiplicative inverse is likewise fairly straightforward: once again using `a` as our additive identity we have y.(x+a) = y.x for all x, y in our field; distributing on the LHS gives y.x + y.a = y.x for all x, y in our field; subtracting y.x from both sides finally gives us y.a = 0 for all y in our field. You would need to relax one or more of the field axioms to have a structure in which the additive identity can have a multiplicative inverse. I'm not aware of any algebraic structure of particular interest that would allow a multiplicative inverse of the additive identity, but in general if you're interested in reading more on this sort of thing I'd recommend reading about rings, commutative rings, and division algebras.