3 ms·
You can have a multiplicative inverse of 0 if 0 = 1. This results in a field with only a single element.
by daveFNbuck 7y ago
You can have a multiplicative inverse of 0 if 0 = 1. This results in a field with only a single element.
- gizmo686 7y agoUnless your definition of field include the non-triviality axiom that 0 != 1.
- Certhas 7y agoNot to be confused with the field with one element :P https://en.wikipedia.org/wiki/Field_with_one_element https://en.wikipedia.org/wiki/Field_with_one_element
- throwawaymath 7y agoThe definition of a field explicitly requires the additive and multiplicative identities to be distinct elements. You can't induce a field structure on a singleton set. You can construct a trivial ring, but the trivial ring does not comprise a trivial field because it fails to be an integral domain. To be fair this is mathematical pedantry when we're talking about trivial objects. But in principle it's meaningful because every field you can construct will lose critical properties if you allow one element to be both the additive and multiplicative identity. Unfortunately authors don't always make it clear that you need both existence and uniqueness when they state the field axioms.