Timeline for Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

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Dec 28, 2012 at 17:50	comment	added	Franz Lemmermeyer		Actually your argument may be completed as follows: if K has odd degree n and H is abelian of degree 2n, let F be the quadratic subfield. This extension is ramified at some prime p, and the ramification must survive when lifted to FK/K. Thus there is no such example.
Dec 28, 2012 at 13:28	comment	added	Chandan Singh Dalawat		All I can say is that while Algebra is of some use, Arithmetic is far too subtle.
Dec 28, 2012 at 11:26	comment	added	Franz Lemmermeyer		If K is an abelian of odd degree over Q, then the 2-class group is never cyclic. This follows easily by looking at the action of the Galois group on the class group. The same thing holds for new-class groups over arbitrary base fields F.
Dec 28, 2012 at 8:41	history	edited	Chandan Singh Dalawat	CC BY-SA 3.0	added 9 characters in body
Dec 28, 2012 at 7:11	history	answered	Chandan Singh Dalawat	CC BY-SA 3.0