Exhibit elements in a quadratic field having the same norm but which are not conjugate or associate

Exhibit elements in a quadratic field which have the same norm but which are not conjugate or associate.


Let K = \mathbb{Q}(i). If (a,b,c) is a Pythagorean triple, then N(a+bi) = N(c), but a+bi and c are clearly neither conjugate nor associate. For instance, \{3+4i,5\} and \{5+12i,13\}.

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