You’ve understood correctly minus one important detail:

The structure you describe (where we want elements and their inverses to have finite support)

Not elements and their inverses! Elements *or* their inverses. I’ve shown the example of to demonstrate that you quickly get infinite inverses, and you’ve come up with an abstract argument why finite inverses won’t cut it:

To show that nothing else works, let and be any two nonzero sums of finitely many integer powers of (so like ). Then, the leading term (product of the highest power terms of and ) will be some nonzero thing. But also, the smallest term (product of the lower power terms of and ) will be some nonzero thing. Moreover, we can’t get either of these to cancel out. So, the product can never be equal to . (Unless both are monomials.)

In particular, your example of has the inverse . Perhaps a better way to describe this set is ‘all you can build in finitely many steps using addition, inverse, and multiplication, starting from only elements with finite support’. Perhaps you can construct infinite-but-periodical elements with infinite-but-periodical inverses; if so, those would be in the field as well (if it’s a field).

If you can construct , it would not be field. But constructing this may be impossible.

I’m currently completely unsure if the resulting structure is a field. If you get a bunch of finite elements, take their infinite-but-periodical inverse, and multiply those inverses, the resulting number has again a finite inverse due to the argument I’ve shown in the previous comment. But if you use addition on one of them, things may go wrong.

A larger structure to take would be formal Laurent series in . These are sums of finitely many negative powers of x and arbitrarily many positive powers of . This set is closed under multiplicative inverses.

Thanks; this is quite similar—although not identical.

Yeah, that’s conclusive. Well done! I guess you can’t divide by zero after all ;)

I think the main mistake I’ve made here is to assume that inverses are unique without questioning it, which of course doesn’t make sense at all if I don’t yet know that the structure is a field.

So, I guess one possibility is that, if we let [x] be the equivalence class of all elements that are =x in this structure, the resulting set of classes is isomorphic to the Laurent numbers. But another possibility could be that it all collapses into a single class—right? At least I don’t yet see a reason why that can’t be the case (though I haven’t given it much thought). You’ve just proven that some elements equal zero, perhaps it’s possible to prove it for all elements.