Algorithm for the class field tower problem?

Question

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?

Answer 1

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.

Answer 2

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.