[Tainnor]

I've never seen anybody allow a field with just a single element, usually it says "a field is a ring with 1!=0...". But yeah, I suppose you could allow it, it's just an incredibly boring ring in which you actually can't divide at all, because there is no nonzero element.

[Tainnor]

eh that was wrong, you can of course divide in the zero ring, because the single element is its own inverse.

[eru]

Yes. Of course, still an incredibly boring field.