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 ...](http://books.google.es/books?id=QEocn1kbfxAC&lpg=PP1&ots=lIgHD9F4wU&dq=Hecke%20lectures&pg=PA107#v=onepage&q&f=false), 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?
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> 


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.



Greatest common divisor of algebraic integers
I give a proof of this result -- actually the stronger result that the GCD of any two algebraic integers may be expressed as a linear combination -- in $\S 23.4$ of [these commutative algebra notes](http://alpha.math.uga.edu/%7Epete/integral.pdf).


The required input from algebraic number theory is nontrivial -- namely that the ideal class group of (the ring of integers of) a number field is finite -- but is much less than that of class field theory.


Note though that one could get away with knowing that these ideal class groups are **torsion** abelian groups, which is, *a priori*, a more structural and thus possibly easier to prove result. I have been racking my brains trying to come up with a more fundamentally commutative algebraic proof of this fact, thus far without success.


