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][1].

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.






[1]: http://math.uga.edu/~pete/integral.pdf