Ring theory MOC

Euclidean domain

A Euclidean domain is an integral domain with a generalized version of the Euclidean division algorithm. More precisely, an integral domain is called a Euclidean domain iff there exists a Euclidean function such that¹ ring

1. for all nonzero ; and
2. if and , then there exist elements such that and .

Every Euclidean domain is a Principal ideal domain.

Proof

proof

Properties

• The polynomial ring over a field is a Euclidean domain

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Footnotes

1. 2017. Contemporary abstract algebra, §18, p. 315. ↩