Let $\langle K, +, * \rangle$ be our field. By definition, we know that every non-zero element, i.e every element except the additive identity, has an multiplicative inverse in the field, and we also do know that every element, including the multiplicative identity, has a additive inverse in the field.
However, giving the fact that additive and multiplicative operations are just binary operations, and they are just represented by different symbols, I would expect both addition and the multiplication to behave in the same way in the sense that if multiplicative identity has an additive inverse, so should the additive inverse to have a multiplicative inverse, i.e a symmetry between them.
So my question is exactly which property of those binary operations or the proper of the field is corresponds to this unsymmetrical behaviour ?
Best Answer
As I said in the comments, the symmetry is broken because of the distributivity, $a(b+c) = ab + ac$, which does not hold symmetrically (indeed $a+bc = (a+b)(a+c)$ is not true in general).
This identity, the group axioms for $+$ and the fact that $0\neq 1$ altogether imply that $0$ has no multiplicative inverse ($0a = (0+0)a = 0a + 0a$ and so $0a=0$, so unless $0=1$, $0$ has no inverse)
These axioms are there to generalize $\mathbb{Z},\mathbb{Q}$ (integral domains, or more generally rings with more than one element), and are therefore "natural", because in particular distributivity is essential in those structures.