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    Euclidean Domains and PIDS

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    Please help with the following problem.

    Show that the ring Z[i] = Z[x] / (x^2 + 1) is a Euclidean Domain, under N(n+mi) = n^2 + m^2. Then, conclude that Z[i] is a Principal Ideal Domain.

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    Solution Preview

    Problem: Show that the ring Z[i] = Z[x] / (x^2 + 1) is a Euclidean Domain, under N(n+mi) = n^2 + m^2. Then, conclude that Z[i] is a Principal Ideal Domain.

    Solution:

    See two attachments for full workings of the solution.

    Consider the multiples of a Gaussian integer z=k+mi. We have
    (k+mi)(l+ni)=(kl-mn)+(kn+lm)i (1)
    Let us consider the following particular cases:
    l=1. In that case the right-hand ...

    Solution Summary

    This solution discusses Euclidean domains and principal ideal domains. Step by step calculations are provided.

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