Notes 101.15 An elementary proof that not all principal ideal domains are Euclidean domains

摘要

A standard result in undergraduate algebra courses is that every Euclidean domain (ED) is a principal ideal domain (PID). It is routinely stated, but rarely proved, that the converse is false. The ring R = L[θ], where θ = 1/2(1 + √-19), so that θ~2 = θ - 5, is sometimes mentioned as is sometimes mentioned as a counterexample. The proof that R is indeed a counterexample is due to Motzkin [1] in 1948, as a special case of much more general results. A more elementary proof, accessible to advanced undergraduates, is given by Cámpoli [2] in 1988, though with a more restricted definition of Euclidean norm than Motzkin uses.