I'm working through Herstein's "Abstract Algebra" text, and am currently working through section 5.
Theorem 4.5.5 introduces the division algorithm for polynomial rings over fields, which states:
Given the polynomial $f(x), g(x) \in F[x]$, where $g(x) \neq 0$, then $$f(x) = q(x)g(x) + r(x),$$ where $q(x), r(x) \in F[x]$ and $r(x) = 0$ or $\deg r(x) < \deg g(x).$
What requirements must be put on a ring to ensure a division algorithm exists? It seems that the existence of some kind of norm is necessary (in this case, the $\deg$ function).
In other words, does the division algorithm hold, say, in any integral domain? Or do you need a unique factorization domain, or perhaps a principle ideal domain instead? My thought is that it almost certainly holds in any Euclidean domain, but I'm wondering if this is too strong of a requirement.