What's an example of two quadratic integers in the same quadratic field for which N(a)|N(b), yet a does not divide b?
Fix a squarefree integer . Then is a quadratic field. So, . With gives a relationship between them. Let such that . Then . There exists such that . If does not divide , then there exists a unique with and (note that , as this would imply divisibility). Use these relationships to find an example.