Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general unramified extensions, is there a bound (depending on $K$) on the degree of an unramified extension over $K$? If so, does the compositum of all unramified extensions also have finite degree over $K$ in general?
Thanks for your attention!
ADDENDUM: as Hunter noticed the answer can be no even just even for solvable groups, when the field admits an infinite class field tower. But perhaps it is still interesting to study the question for extension having simple Galois group, and possibly their compositum. Is there anything known about this case?
Best Answer
No- even the process of iteratively taking the Hilbert class field, the Hilbert class field of the Hilbert class field, etc, need not terminate. See
http://en.wikipedia.org/wiki/Golod-Shafarevich_theorem