MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?

share|cite|improve this question
Nice question! A possible candidate: the omnific integers or some other ring associated closely with the ordinals. The ordinals themselves support division in the form $\forall\alpha,\beta\exists\gamma,r\ \beta=\alpha\gamma+r$, where $r\lt\alpha$, since one takes as many copies of $\alpha$ that fit into $\beta$, and $r$ is the leftover part. I'm not sure, however, if this feature extends fully to the Omnific integers with the symmetric arithmetic operations of the surreal numbers. – Joel David Hamkins Oct 21 '12 at 2:18
Or the polynomials over a field, but allowing transfinite ordinal degrees, which add by symmetric ordinal addition. – Joel David Hamkins Oct 21 '12 at 3:01
As the floor function is defined in the surreals (floor(x)=\{x−1|x+1\}) is an omnific integer, Euclidian division is indeed possible – Feldmann Denis Oct 21 '12 at 3:30
I don't think the omnific integers can admit an ordinal-valued (as opposed to omnific integer-valued) Euclidean function; an ordinal-valued Euclidean function should still lead to being a unique factorization domain, in the usual way, but the omnific integers lack unique factorization (e.g., 2, 3, 5, 7, ..., all remain prime, but $\omega$ is divisible by all infinitely many of them). – Sridhar Ramesh Oct 21 '12 at 18:53

Yes, they exist. Even if the problem is left open in the the papers of T. Motzkin and P. Samuel cited in Comparing different Euclidean algorithms on a Euclidean domain

the problem is solved in

J. Hiblot, Des Anneaux euclidiens dont le plus petit algorithms n'est pas valeurs finies, C. R. Acad. Sci. Paris, 281 (1975), 411-414 with correction in vol 287.

M. Nagata, On Euclid algorithm. C. P. Ramanujama tribute, pp. 175 Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978.

See also:

Pag. 195 in P. M. Cohn, Free Ideal Rings and Localization in General Rings

Rod Downey, Euclidean Domains and Euclidean Functions,

Pete L. Clark, A note on euclidean order types,

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.