Gcd not implies Bezout

This article gives the statement and possibly, proof, of a non-implication relation between two integral domain properties. That is, it states that every integral domain satisfying the first integral domain property need not satisfy the second integral domain property
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Statement

There exist gcd domains which are not Bezout domains.

Proof

Since every UFD is a gcd domain, it suffices to construct an example of a UFD which is not a Bezout domain. In particular, it suffices to construct an example of a Noetherian UFD which is not a PID. One such example is the multivariate polynomial ring over a field, i.e. the polynomial ring over a field, in more than one variable.