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Let $k$ be a number field. Is it possible that $k$ has an infinite (non-abelian) extension that is unramified everywhere?

Thank you!

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See also $$ $$ mathoverflow.net/questions/58047/… $$ $$ –  Chandan Singh Dalawat Sep 28 '11 at 10:23
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up vote 12 down vote accepted

Two things:

1) Yes, certainly. By class field theory and the finiteness of the class group, the maximal abelian unramified extension of any number field is of finite degree. Thus any infinite unramified extension is non-abelian -- in particular, any infinite class field tower. The Golod-Shafarevich examples and oodles of refinements since then all give examples.

2) Perhaps you are actually interested in non-solvable infinite extensions, i.e., unramified extensions which are not built up as an infinite series of unramified abelian extensions. In this case, the answer is also yes. In fact, this can even be done over number fields of class number 1. For examples, see Maire's "On Infinite Unramified Extensions."

Hope that helps.

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