Algorithm for the class field tower problem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:16:58Z http://mathoverflow.net/feeds/question/22302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22302/algorithm-for-the-class-field-tower-problem Algorithm for the class field tower problem? Pete L. Clark 2010-04-23T05:33:00Z 2010-04-23T08:19:55Z <p>This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:</p> <p>Let $K$ be a number field. The (Hilbert) <strong>class field tower</strong> 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.</p> <p>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?</p> http://mathoverflow.net/questions/22302/algorithm-for-the-class-field-tower-problem/22306#22306 Answer by Cam McLeman for Algorithm for the class field tower problem? Cam McLeman 2010-04-23T06:20:29Z 2010-04-23T07:54:55Z <p>Not in the slightest! The answer is not even known for quadratic imaginary number fields. In fact, the <em>only</em> 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).</p> <p>Incidentally, the "sufficiently many primes ramified" business is a bit of a red herring, in my opinion. The <i>real</i> 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.</p> http://mathoverflow.net/questions/22302/algorithm-for-the-class-field-tower-problem/22315#22315 Answer by Franz Lemmermeyer for Algorithm for the class field tower problem? Franz Lemmermeyer 2010-04-23T08:19:55Z 2010-04-23T08:19:55Z <p>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.</p> <p>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</p> <ul> <li>F. Hajir, <em>On the growth of $p$-class groups in $p$-class field towers</em>,<br> J. Algebra 188, No.1, 256-271 (1997)</li> </ul> <p>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. </p> <ul> <li>H. Nover, <em>Computation of Galois groups associated to the 2-class towers of some imaginary quadratic fields with 2-class group $C_2 \times C_2\times C_2$</em>, J. Number Theory 129, No. 1, 231-245 (2009)</li> </ul> <p>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.</p>