2 added reference to dror comment

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 in $I$ 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 $K$ (though I haven't seen EDIT: see Dror's comment for an illustrative 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?

1

# Where does the principal ideal theorem (from CFT) go?

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 in $I$ 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 $K$ (though I haven't seen an illustrative 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?