Question

Show that every Euclidean domain is a principal ideal domain.

Answer 1

Let (D,+,×)(D,+,×) be a Euclidean domain whose zero is 00 and whose Euclidean valuation is νν.

We need to show that every ideal of (D,+,×)(D,+,×) is a principal ideal.

Let UU be an ideal of (D,+,×)(D,+,×) such that U≠{0}U≠{0}.

Let d∈Ud∈U such that d≠0d≠0 and ν(d)ν(d) is as small as possible for elements of UU.

By definition, νν is defined as ν:D∖{0R}→Nν:D∖{0R}→N, so the codomain of νν is a subset of the natural numbers.

By the Well-Ordering Principle, such an element dd exists as an element of the preimage of the least member of the image of UU.

Let a∈Ua∈U.

Let us write a=dq+ra=dq+r where either r=0r=0 or ν(r)<ν(d)ν(r)<ν(d).

Then r=a−dqr=a−dq and so r∈Ur∈U.

Suppose r≠0r≠0.

That would mean ν(r)<ν(d)ν(r)<ν(d) contradicting dd as the element of UU with the smallest νν.

So r=0r=0, which means a=qda=qd.

That is, every element of UU is a multiple of dd.

So UU is the principal ideal generated by dd.

This deduction holds for all ideals of DD.

Hence the result.

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