Why Hurwitz's lemma(Hurwitz's lemma says that there exists a positive integer M with the following properties that for a,b(nonzero)belongs to ring of integers there is a t, 1<=t<=M & w belongs to ring of integers such that |N(at-bw)|<|N(b)|)is called as a weak generalization of the Euclidean Algorithm? Can some one give an example where Euclidean Algorithm fails true but this lemma holds true? I'm thinking about in Z[√-5] can you elaborate give me an answer in this ring of integers? It's an important question from Ireland & rosen book " A Classical Introduction to Modern Number theory".
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2Can you please expand your post to make it understandable, and add some context? https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question – Anne Bauval Apr 23 '23 at 06:30
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See e.g. the papers by Lemmermeyer and Fendel that I cite in this comment – Bill Dubuque Apr 23 '23 at 07:02