23. Let Z[−1]={α=a+b−1:a,b∈Z},N(α)=a2+b2. (i) Introduce the concepts of divisibility, irreducible elements, and prime elements in Z[−1], and establish the basic properties analogous to the first part of Chapter 1, Section 2. (ii) Let α,β∈Z[−1]. If α∣β and β∣α hold simultaneously, then α,β are called associates. Prove that α,β are associates if and only if α=±β,±iβ.
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Official solution
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Source: NuminaMath-1.5,
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