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.)