Let be an irreducible Gaussian integer with . Show that if is a factor of its conjugate , then is an associate of .
Let and suppose . This equality yields the two equations and , which can be rearranged as and . Now , so that $latex . Hence .
If , then , so that , a contradiction. Similarly, if then and we have . If , then , so that . If , then , and we have . In either case, . If , then has a nontrivial factorization (namely ) and so is not irreducible. Thus , and so . These are precisely the associates of .