Yes, I should have been more precise: "modulo some power of two, which is not a prime, unless we're talking about 2^1".

Now I'm not aware of any language where there is an integer type that has only two elements, unless you're talking about booleans - but booleans are so special that, while they're of course isomorphic to GF(2), we don't really use the same words (addition and multiplication) but different ones (exclusive disjunction (xor) and conjunction (and)).

So, in any real-life situation, your fixed-size integers won't form a field, because the ring of integers modulo n is only a field when n is prime, and so in particular, division by nonzero elements will be undefined in general.

And yes, of course, you can't divide by zero, but that's also true of the real numbers themselves, so no surprises there...

(I guess a better point would be that division is mathematically "broken" for integers anyway, since integers technically also don't form a field and, depending on the language, you may either get back a truncated result from division or will get a different type (ratio or floating point).)

> Yes, I should have been more precise: "modulo some power of two, which is not a prime, unless we're talking about 2^1".

If you want to be even more nit-picky, 2^0 might also work. But whether your definition of a field admits fields with only a single element is just a question of taste, that is even less important or deep than whether your flavour of natural numbers includes 0 or not.

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.

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

Yes. Of course, still an incredibly boring field.

Has anyone yet demonstrated a field with a single element?

It's not a question of demonstration but of definition. I've only ever people define fields as requiring two distinct elements 0 and 1. However, every field is also a ring and there can be, up to isomorphism, only a single ring with one element, because you don't really get any choice as to how you define the operations. That's called the trivial ring or zero ring and it basically satisfies all of the axioms for a field too, except for not having two distinct elements. So if it were possible for a field with a single element to exist, it would have to be this one.

It is my understanding that some mathematicians are looking into some objects that are kind of "like" a field with one element, but not in the sense of classical algebra, see: https://en.m.wikipedia.org/wiki/Field_with_one_element

However, I haven't really looked into that.