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This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:

Let $K$ be a number field. The (Hilbert) class field tower of $K$ is the sequence defined by $K^0 = K$ and for all $n \geq 0$, $K^{n+1}$ is the Hilbert class field of $K^n$. Put $K^{\infty} = \bigcup_n K^n$. We say that the class field tower is infinite if $[K^{\infty}:K] = \infty$ (equivalently $K^{n+1} \supsetneq K^n$ for all $n$). Golod and Shafarevich gave examples of number fields with infinite class field tower, and thus which admit everywhere unramified extensions of infinite degree. It is now known that a number field with "sufficiently many ramified primes" has infinite class field tower.

My question is this: is there a known algorithm which, upon being given a number field, decides whether the Hilbert class field tower of $K$ is infinite?

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up vote 9 down vote accepted

Not in the slightest! The answer is not even known for quadratic imaginary number fields. In fact, the only known way to show that the Hilbert class field tower of a number field is infinite is to invoke one of a variety of different forms of Golod-Shafarevich, and I don't think it's even seriously conjectured (more like "wondered") that every infinite Hilbert class field tower arises by applying Golod-Shafarevich to some step in the tower (or to some cleverly chosen subfield).

Incidentally, the "sufficiently many primes ramified" business is a bit of a red herring, in my opinion. The real condition is that the $p$-rank of the class group is large for some prime $p$. When $K$ is cyclic of degree $p$, it is only the fact that genus theory relates the $p$-rank of the class group to the number of ramified primes that brings ramified primes into the picture. (For example, the standard Golod-Sharevich examples come from showing the 2-class field tower is infinite by using Gauss' result that many primes ramifying in a quadratic extension imply a large 2-rank). For non-cyclic extensions, the link is more tenuous, and it becomes much more natural to talk strictly in terms of the class group.

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There's little if nothing to add to Cam's answer, except that I want to point out that there is a big technical difference between class field towers and $p$-class field towers. I have never seen any conjecture in the direction of the statement "if $K$ has infinite class field tower, then some subfield of the class field tower has infinite $p$-class field tower for some prime $p$". All known infinite class field towers in fact come from some $p$-class field tower, for which Golod-Shafarevich applies.

Thus general class field towers are a very difficult topic. For $p$-class field towers, on the other hand, I would guess that most specialists indeed think that if such a tower is infinite, then some subfield satisfies the Golod-Shafarevich bound. In this connection, see

  • F. Hajir, On the growth of $p$-class groups in $p$-class field towers,
    J. Algebra 188, No.1, 256-271 (1997)

But even if this were known, there would not be a terminating algorithm for deciding the finiteness of the $p$-class field tower. There are nontrivial cases in which the $2$-tower was shown to be finite; for some recent calculations see e.g.

  • H. Nover, Computation of Galois groups associated to the 2-class towers of some imaginary quadratic fields with 2-class group $C_2 \times C_2\times C_2$, J. Number Theory 129, No. 1, 231-245 (2009)

This approach shows that certain types of class groups in small subfields prevent the $p$-class field tower from becoming infinite for group theoretic reasons. But there's a large gap between these results and Golod-Shafarevich, where no one really knows what is happening.

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I'm intrigued by the last comment -- "where no one really knows what is happening." Not that I'm disagreeing, but did you have something specific in mind here? –  Cam McLeman Apr 23 '10 at 14:49
    
I'm not sure I understand your question; what I meant to say was that the p-tower is finite if the class groups are "small", and that it is infinite if they are "large", and that neither group theoretical calculations nor improvements of Golod-Shafarevich tell us what to expect in the cases in between. Personally I fear that there is no nice and simple result that separates the finite towers from the infinite. –  Franz Lemmermeyer Apr 23 '10 at 18:13
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