Content deleted Content added
→History: Red link Tags: Mobile edit Mobile web edit Disambiguation links added |
m Undid revision 1222707540 by Paulinefowler666 (talk) ??? |
||
Line 29:
== Expected properties ==
=== '''F'''<sub>1</sub> is not a field ===
'''F'''<sub>1</sub> cannot be a field because by definition all fields must contain two distinct elements, the [[additive identity]] zero and the [[multiplicative identity]] one. Even if this restriction is dropped (for instance by letting the additive and multiplicative identities be the same element), a ring with one element must be the [[zero ring]], which does not behave like a finite field. For instance, all [[Module (mathematics)|modules]] over the zero ring are isomorphic (as the only element of such a module is the zero element). However, one of the key motivations of '''F'''<sub>1</sub> is the description of sets as "'''F'''<sub>1</sub>{{nbh}}vector spaces" – if finite sets were modules over the zero ring, then every finite set would be the same size, which is not the case. Moreover, the [[Spectrum of a ring|spectrum]] of the trivial ring is empty, but the spectrum of a field has one point.
| |||