Conductors of non-abelian number fields? Is there a definition out there of the notion of conductor of a non-abelian number field (i.e. a finite extension of Q whose Galois group is non-abelian)? If not, is there anyone you know of working on it? The definition for abelian number fields uses class field theory; it comes out of Artin reciprocity (see page 525 of Neukirch's Algebraic number theory).
 A: I think that a good notion of "conductor" isn't going to be intrinsic to the extension K/Q; rather, you might choose some finite-dimensional complex representation rho of Gal(K/Q) and then use the Artin conductor of the resulting Galois representation.  When K/Q is abelian, there aren't so many interesting choices of rho.
A: I don't know the precise definition of the conductor of a nonabelian Galois extension of Q, but see page 10 of Langlands's expository article "Representation Theory: Its Rise and Its Role in Number Theory" sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/gibbs-ps.pdf . Presumably the thesis of Joe Buhler referenced therein gives a precise definition, or a reference to one.
A: The conductor of an order O \subset K is defined on p. 79 of Neukirch's Algebraic Number Theory (so that for K you define the conductor to be the conductor of its ring of integers). This conductor tells you how to compute how a prime decomposes in K. I think in the abelian case these are the same conductors as in class field theory.
In the non-abelian case, maybe the conductor is related to the zeta function of K (I'm not sure off the top of my head).
