My impression is that one of the celebrated results of class field theory the principal ideal theorem namely that given a number field $K$ and its maximum unramified abelian extension L, every ideal in the ring of integers $O_K$ of $K$ becomes principal in the ring of integers $O_L$ of $L$. That is, given an ideal $I$ in $O_K$, the ideal $I \dot O_L$ is principal.

This result was originally conjectured by Hilbert in 1900 and reduced to a group theoretic question by Artin which was finally solved by Furtwangler in 1930.

I've never seen any further discussion of the principal ideal theorem - I don't know any generalizations or applications.

As James Milne comments in Remark 3.20 of the fifth chapter of his book on class field theory it's easy to see that there is *some* finite extension of $K$ for which all ideals of $K$ become principal. He further comments that this extension need not be the Hilbert class field of $K$ (EDIT: see Dror's comment for an example).

Is the principal ideal theorem primarily of historical interest (e.g. because it was a long standing conjecture of Hilbert)? Or does it have some deeper significance?

somenumber field with class number 1 iff the class field tower over K is a finite extension of K, in which case the top field in the tower is an example of a finite extension of K with class number 1. The proof of this iff statement doesnotrequire the principal ideal theorem, although that theorem doesmotivatethe result! $\endgroup$ – KConrad Apr 30 '11 at 4:37