ABSTRACT
The difficulty we encountered in the last chapter in proving that Z[x] is a UFD might convince you of the advantage of having a Division Theorem available: The Division Theorem makes proving that Z and Q[x] are UFDs relatively easy. It seems natural then to define a more general class of domains (including both Z and Q[x]) that have a Division Theorem. We name this class of domains in honor of Euclid, in whose Elements we find the first reference to that corollary of the Division Theorem, Euclid’s Algorithm. This is a typical gambit of mathematicians. We have identified an important tool we’d like to study in general (in this case, a Division Theorem for domains), and so we isolate those domains having this tool by means of a definition.
