> Every element EXCEPT 0 has a multiplicative inverse, a⁻, such that a*a⁻ = 1.
What is "0"? It's not defined in the axioms other than additive zero. Or is it multiplicative zero? (1?). Is it the number zero?
If it is the additive zero defined in axiom (3), then it just seems weird to me that additive zero is undefined for multiplicative inverse for all fields always and forever.
If it is the number zero, then how does that generalize to other fields?
If the answer is "Numbers are the first field and all fields generalize that", then I suppose we are referring to the number (0), and that's fine, as other fields are welcome to define their own larger definition of zero that includes the number (0) ... ?
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.
"Zero" is just a specific element of the field which satisfies being the additive identity as well as the rest of the properties of a field where 0 is mentioned. When the rest of the axioms refer to "zero" they refer to the exact same element of the set that is also the additive identity.
It's not the "number zero" because a field does not care about numbers, it's just elements of a set (which might be numbers like in R's case).
1 is not "multiplicative zero", it's the "multiplicative identity".
0 and 1 are just the shorthand we give for those elements. because those are the symbols we use in R which is the most common field we deal with in everyday life.
What is "0"? It's not defined in the axioms other than additive zero. Or is it multiplicative zero? (1?). Is it the number zero?
If it is the additive zero defined in axiom (3), then it just seems weird to me that additive zero is undefined for multiplicative inverse for all fields always and forever.
If it is the number zero, then how does that generalize to other fields?
If the answer is "Numbers are the first field and all fields generalize that", then I suppose we are referring to the number (0), and that's fine, as other fields are welcome to define their own larger definition of zero that includes the number (0) ... ?