Proposition 1

Let $K$ be the field of fractions of a principal ideal domain $A$, and let $(x_i)_{i \in I}$ be a family of elements of $K$ having a common denominator $b \in K^*$ (in other words $bx_i \in A$ for all $i$). Then:

a) The family $(x_i)$ has a gcd in $K$. b) Every gcd of $(x_i)$ can be expressed in the form $d = \sum a_i x_i$, where the $a_i$ are elements of $A$, all but finitely many of which are zero.

Indeed the ideal $\sum Abx$, of $A$ is principal, and so of the form $Ad'$. Put $d' = bd \ (d \in K)$. From the relation $d' = \sum a_i b x_i$, we deduce $d = \sum a_i x_i$, where $a_i \in A$. Hence every common divisor of the $x_i$ divides $d$. On the other hand, since $bd$ is a common divisor of the $bx$, by construction, it follows that $d$ is a common divisor of the $x_i$.