Formally we say that a ring is a Euclidean domain if:
- It is an integral domain.
- There a function called a Norm such that for all nonzero there are such that and either or .
Some common examples of Euclidean domains are:
- The ring of integers with norm given by .
- The ring of Gaussian integers with norm given by .
- The ring of polynomials over any field with norm given by .
It can be easily shown through infinite descent that any Euclidian domain is also a principal ideal domain. Indeed, let be any ideal of a Euclidean domain and take some with minimal norm. We claim that . Clearly , because is an ideal. Now assume and consider any . Applying the division algorithm we get that there are such that with (we cannot have as ). But now as is an ideal, and , we must have , contradicting the minimality of . Hence and is indeed a principle ideal domain.
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