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Is there a good way to show that the Gaussian integers are a Principal Ideal Domain without using the fact that they are a Euclidean Domain? It seems like a lot of extra structure to need to prove along the way.

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    $\begingroup$ It's the usual way to prove the integers are a PID, too, so I don't know that I'd expect anything different for the Gaussians. $\endgroup$ – Gerry Myerson Oct 1 '15 at 7:21
  • $\begingroup$ You could instead compute the class number using the usual techniques, but that seems like more work to me. $\endgroup$ – Qiaochu Yuan Oct 1 '15 at 19:34

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