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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?

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The smallest illustrative example is probably the following: $\mathbb{Q}(\sqrt{-5})$ has class number two, generated by the ideal above the ramified prime $2$. Taking, as Milne suggests, the square root of $2$ makes for an extension as required, but this is not the Hilbert class field, and is in fact ramified at 2. The Hilbert class field is $\mathbb{Q}(\sqrt{-5},i)$. –  Dror Speiser Apr 29 '11 at 20:36
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Consider the sequence of fields K_0, K_1, K_2, ... where K_0 = K and, for i > 0, K_i is the Hilbert class field of K_{i-1}. These fields are called the class field tower over K, since each K_i is in K_{i+1}. One can show that K can be embedded in some number 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 does not require the principal ideal theorem, although that theorem does motivate the result! –  KConrad Apr 30 '11 at 4:37

3 Answers 3

up vote 12 down vote accepted

Among the generalizations that I can recall off the top of my head are:

  • the generalization to ray class groups already mentioned by Kevin, proved by Iyanaga pretty much immediately after Furtwängler's proof;
  • Furtwängler's own theorem saying that if the class group is an elementary abelian $2$-group, then its basis can be chosen in such a way that each basis element capitulates in some quadratic extension;
  • the theorem of Tannaka and Terada, according to which ambiguous classes in cyclic extension already capitulate in the genus field (the obvious generalization to central extensions fails at least group theoretically due to results of Miyake)
  • the theorem of Suzuki, which claims that in any abelian unramified extension $L/K$, a subgroup of order $\ge (L:K)$ must capitulate; this was generalized by Gruenberg and
    Weiss (Capitulation and transfer kernels).

Capitulation is also at the center of the Greenberg conjecture in Iwasawa theory. In addition, its analogue in the theory of abelian varieties is the visibility of Tate-Shafarevich groups; for a recent contribution in the other direction see Schoof and Washington's article on´the "Visibility of ideal classes".

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There is a generalization of the principal ideal theorem to ray class groups:

Let $K$ be a number field, $\mathfrak{m}$ a modulus for $K$, $L:=K(\mathfrak{m})$ the ray class field modulo $\mathfrak{m}$, and $\mathfrak{n}$ the image of $\mathfrak{m}$ in $L$. Then any element in the ray class group modulo $\mathfrak{m}$ of $K$ becomes trivial in the ray class group modulo $\mathfrak{n}$ of L.

The proof is almost identical to that of PIT, reducing to the same group theoretic statement.

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The general phenomenon, of ideals becoming principal in an extension, is called "capitulation". There has been work going on for a century now. Some results are mentioned in this grant report I found: http://gow.epsrc.ac.uk/ViewGrant.aspx?GrantRef=EP/C517903/1 .

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