So here's something I stumbled across today.
So, the standard definition of a Euclidean domain requires that the Euclidean norm be integer-valued, right? But obviously if you allowed it to take values in a general well-ordered set, you'd get all the same results (IMO we should define Euclidean domain this way but we don't :P ). But, we can define this, call it a transfinite Euclidean domain. Then we can define its Euclidean rank to be the smallest ordinal you can have its Euclidean norm take values in.
Then the question is, is this actually a nontrivial notion? Can you find a transfinite Euclidean domain that is not just a Euclidean domain (i.e. with Euclidean rank greater than ω)?
Apparently the answer is "yes", and this has been known since the 70s. But the methods for this only got domains with Euclidean rank ω².
(First I'm learning of this. I'd been wondering about this since I was, like, a PROMYS student, and I'd never heard of these examples till now.)
But anyway the point is I came across this paper, which goes further and shows that you can find transfinite Euclidean rings of Euclidean rank ωα for any ordinal α. (And these are the only possibilities.) So, that's pretty neat! I haven't read the proof or anything but still yay question answered.
So, the standard definition of a Euclidean domain requires that the Euclidean norm be integer-valued, right? But obviously if you allowed it to take values in a general well-ordered set, you'd get all the same results (IMO we should define Euclidean domain this way but we don't :P ). But, we can define this, call it a transfinite Euclidean domain. Then we can define its Euclidean rank to be the smallest ordinal you can have its Euclidean norm take values in.
Then the question is, is this actually a nontrivial notion? Can you find a transfinite Euclidean domain that is not just a Euclidean domain (i.e. with Euclidean rank greater than ω)?
Apparently the answer is "yes", and this has been known since the 70s. But the methods for this only got domains with Euclidean rank ω².
(First I'm learning of this. I'd been wondering about this since I was, like, a PROMYS student, and I'd never heard of these examples till now.)
But anyway the point is I came across this paper, which goes further and shows that you can find transfinite Euclidean rings of Euclidean rank ωα for any ordinal α. (And these are the only possibilities.) So, that's pretty neat! I haven't read the proof or anything but still yay question answered.
