I have several questions concerning some properties of algebraic numbers. The first concerns the folowing statement:

*Given algebraic integers $\alpha$ and $\beta$ they have a unique greatest common divisor modulo asociates. ie there is an algebraic integer $\delta$ with $\delta \vert \alpha$ and $\delta \vert \beta$ and such that for any other integer $\gamma$ such that $\gamma \vert \alpha$ and $\gamma \vert \beta$ we also have $\gamma \vert \delta$; any other algebraic integer with the same properties is an associate of $\delta$.*

This result has a simple proof using class field theory, ie if $H$ is the Hilbert class field of $\mathbb{Q}[\alpha,\beta]$, then by the principal ideal theorem, the ideal $(\alpha,\beta)$ becomes principal in $H$ say $\alpha {\mathcal O}_H + \beta {\mathcal O}_H = \delta {\mathcal O}_H$.

I have been looking for a simpler proof in several books in the subject, the nearest I found is theorem 98 in Hecke's Lectures ..., but I think it is not enough: it finds an integer $A$ such that the set of multiples of $A$ in $\mathbb Q[\alpha,\beta]$ coincides with the ideal $(\alpha,\beta)$ but it does not follow that the same is true in the bigger field $\mathbb Q[\alpha,\beta,A]$. So my question is:

Is there a proof of the statement not using class field theory?

My "ideal" proof will only use elementary properties of algebraic numbers so the statement could be proved just after the introduction of algebraic integers and units in a classical introduction to the subject, but I fear I'm asking too much.