proof that a Euclidean domain is a PID
Let be a Euclidean domain, and let be a nonzero ideal. We show that is principal. Let
be the set of Euclidean valuations of the non-zero elements of . Since is a non-empty set of non-negative integers, it has a minimum . Choose such that . Claim that . Clearly . To see the reverse inclusion, choose . Since is a Euclidean domain, there exist elements such that
with or . Since and is minimal in , we must have . Thus and .
|Title||proof that a Euclidean domain is a PID|
|Date of creation||2013-03-22 12:43:11|
|Last modified on||2013-03-22 12:43:11|
|Last modified by||rm50 (10146)|