Timeline for Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?
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| Feb 26, 2010 at 4:43 | comment | added | Quetzalcoatl | Thanks! explicit finite extensions was what I was looking for. In skimming the paper, it looks very well-written and with only elementary methods as you mentioned. jtnb.cedram.org/item?id=JTNB_1997__9_1_51_0 | |
| Feb 25, 2010 at 13:13 | comment | added | user1073 | In his paper "Unramified Quaternion Extensions of Quadratic Number Fields", Lemmermeyer constructs (using only elementary methods) unramified extensions with the group H_8 of quaternions as the Galois group. He is very explicit about his methods, so you should be able to use his ideas to generate a few easy to remember examples. | |
| Feb 25, 2010 at 3:11 | comment | added | Emerton | There are the famous Golod--Shafarevich towers. | |
| Feb 25, 2010 at 2:18 | comment | added | Cam McLeman | Yes. Of particular interest is Maire's paper "On Infinite Unramified Extensions", in which he explicitly constructs infinite unramified extensions over fields with trivial (or near-trivial) Hilbert class field. Such extensions are necessarily nonb-abelian. | |
| Feb 25, 2010 at 1:12 | history | answered | Quetzalcoatl | CC BY-SA 2.5 |