Any Euclidean domain satisfies the division “algorithm”:
For any $x,d$ there exists $q,r$ such that $x = qd+r$ with $\sigma(r)<\sigma(d)$ or $\sigma(r)=0$
With $\sigma$ a “size function.”
I’m wondering if what I would call an algorithm (i.e. a discrete series of steps to get to an answer) exists for division. Specifically:
Suppose +, -, and * are defined in some Euclidean Domain. Is there a mechanism to find $x/y$ (beyond brute force)?
Repeated subtraction works in the integers, but not the polynomials, and this was my first thought. (I realize that I’m ignoring the problem of how to do subtraction in the ring, which is quite similar.)
Various complexity results are known about Euclidean domains. For example, Downey and Kach: Euclidean functions of computable Euclidean domains shows that there is a computable Euclidean domain having no (finite or transfinitely-valued) computable Euclidean function. Further there is a computable Euclidean domain with a computable Euclidean function but whose units are noncomputable, and there is a computable Euclidean domain with neither computable units nor a computable Euclidean function. See the paper and its references for much more.