Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ actually exists. Assume for simplicity, as Hilbert did, that $K$ is totally complex with class number $2$, and that the ideal class of ${\mathfrak a}$ generates the class group.

We know that the Hilbert class field $L = K(\sqrt{\alpha})$ satisfies $\alpha = \eta \beta$ for some unit $\eta$, where $(\beta) = {\mathfrak a}^2$. The most natural construction would therefore show that among the elements in the "Selmer group" $S$ generated by units and squares of ideals there is an element congruent to a square modulo $4$, but apparently this does not work as directly as we would wish, and Hilbert had to enlarge $S$ by elements which had no chance of generating the class field but without which his proof does not work (he made $S$ so large that two elements had to lie in the same residue class mod $4$, used Dirichlet's box principle, and then showed that the resulting element does not involve any elements outside of $S$).

My question is whether there are other ways of showing the existence of such an $\alpha$ that have at least a chance of working. To put it another way: if you didn't know class field theory, where would you look for a proof of the existence of $\alpha$?