We often want the field without zero to form a multiplicative group, and this isn’t the case in the ring with one element (because the empty set lacks an identity and hence isn’t a group). Indeed we could take the definition of a field to be
A ring such that the non-zero elements form a multiplicative group.
We often want the field without zero to form a multiplicative group, and this isn’t the case in the ring with one element (because the empty set lacks an identity and hence isn’t a group). Indeed we could take the definition of a field to be
and this is fairly elegant.