Thread: Euclidean Domains

Let be a Euclidean Domain. Let be the minimum integer in the set of norms of nonzero elements of . Prove that every nonzero element of norm is a unit in . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.

Originally Posted by dori1123

Let be a Euclidean Domain. Let be the minimum integer in the set of norms of nonzero elements of . Prove that every nonzero element of norm is a unit in . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.

A norm is a function such that for any , we can write where: and , or . (There might be an additional property to the norm function but not all books define it).

Say that where . This means we can write, where or else . But is minimal and so . Thus, which means that . Thus, is a unit by definition. It follows that if then since has the smallest possible norm it means it has to be a unit.