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Is/should there be a theory of finite solvable extensions over a given base field? Could it be based on/use class field theory? Assume the base field isn't a local field.

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Is there any reason to be interested in the maximal solvable extensions? – Ho Chung Siu Nov 6 '09 at 13:47
When Taylor was proving potential modularity a few years ago, he invoked a theorem of Moret-Bailly saying basically that a quasiprojective geom connected alg variety over Q with a Q_p-point for all p in S (finite) had a Q_S-point, where Q_S was the max extension of Q in which all primes in S split. He remarked at the time that if one could get away with the maximal soluble extension of Q with this property then he'd be able to get much much further (resolve Serre's conj etc). – Kevin Buzzard Nov 6 '09 at 15:46
up vote 10 down vote accepted

As FC says, since solvable extensions are built up out of abelian extensions, class field theory is certainly relevant and helpful in understanding the structure of solvable extensions. On the other hand, to do this in a systematic way requires understanding class field theory of each number field in a tower "all at once". The picture that one gets in this way seems quite blurry compared to the classical goal of class field theory: to describe and parameterize the finite abelian extensions L of a field K in terms of data constructed from K itself. In the case of a number field, this description is in terms of groups of (generalized) ideal classes, or alternately in terms of quotients of the idele class group. I'm pretty sure there's no description like this for solvable extensions of any number field.

What I can offer is a bunch of remarks:

1) Sometimes one has a good understanding of the entire absolute Galois group of a field K, in which case one gets a good understanding of its maximal (pro-)solvable quotient. Of course this happens if the absolute Galois group is abelian.

2) Despite the OP's desire to exclude local fields, this is one of the success stories: the full absolute Galois group of a $p$-adic field is a topologically finitely presented prosolvable group with explicitly known generators and relations.

3) On the other hand, we seem very far away from an explicit description of the maximal solvable extension of Q. For instance, in the paper

MR1924570 (2003h:11135) Anderson, Greg W.(1-MN-SM) Kronecker-Weber plus epsilon. (English summary) Duke Math. J. 114 (2002), no. 3, 439--475.

the author determines the Galois group of the extension of Q^{ab} which is obtained by taking the compositum of all quadratic extensions K/Q^{ab} such that K/Q is Galois. Last week I heard a talk by Amanda Beeson of Williams College, who is working hard to extend Anderson's result to imaginary quadratic fields.

4) This question seems to be mostly orthogonal to the "standard" conjectural generalizations of class field theory, namely the Langlands Conjectures, which concern finite dimensional complex representations of the absolute Galois group.

5) A lot of people are interested in points on algebraic varieties over the maximal solvable extension Q^{solv} of Q. The field arithmeticians in particular have a folklore conjecture that Q^{solv} is Pseudo Algebraically Closed (PAC), which means that every absolutely irreducible variety over that field has a rational point. This would have applications to things like the Inverse Galois Problem and the Fontaine-Mazur Conjecture (if that is still open!). Whether an explicit description of Q^{solv}/Q would be so helpful in these endeavors seems debatable. I have a paper on abelian points on algebraic varieties, in which the input from classfield theory is minimal.

The two papers on solvable points that I know of (and very much admire) are:

MR2057289 (2005f:14044) Pál, Ambrus Solvable points on projective algebraic curves. Canad. J. Math. 56 (2004), no. 3, 612--637.

MR2412044 (2009m:11092) Çiperiani, Mirela; Wiles, Andrew Solvable points on genus one curves. Duke Math. J. 142 (2008), no. 3, 381--464.

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Edited. Pete, it's usually considered internet courtesy not to do such things. – Qiaochu Yuan Nov 6 '09 at 15:12
Apologies. I should have known better. – Pete L. Clark Nov 6 '09 at 15:14
Thanks a lot for this interesting post – Jonah Sinick Nov 6 '09 at 15:22
I am curious what evidence there is to support the conjecture that Q^solv is PAC. Also, what about the corresponding statement for F_p(t)? Very interesting post. – Brandon Levin Nov 18 '09 at 23:15
This is not an answer but a small correction to point 2) in Pete Clark's answer. I would have posted it as a comment if I had enough reputation. The absolute Galois groups of local fields of prime characteristic $p$ are not finitely generated. For example, $K=\mathbf{F}_p((t))$ has infinitely many cyclic extensions of degree $p$ (because the $\mathbf{F}_p$-space $K^+/\wp(K^+)$ is not finite-dimensional). – user82218 May 13 at 4:07

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