It turns out that to prove is a Principal Ideal Domain, it is easier to prove that it is a Euclidean domain, and hence a PID.

(Any readers who have a direct proof that is a PID, please comment below, as it would be very interesting to know such a proof. 🙂 )

Proof:

As mentioned above, we will prove that it is a Euclidean domain.

Let .

We need to show: such that , with .

Consider . Define where are the integers closest to respectively.

Then, , where .

.

Take .

Check out recommended Abstract Algebra books: Recommended Books for Math Undergraduates

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