algebraic number theory - Conjugacy classes in the absolute galois group

I agree with the answer of Kevin Buzzard -- you better look only at quotients $Gal(K/Q)$ unramified outside some (finite) set $S$ to make the question make sense as it is.

But regardless, you can ask about the conjugacy classes in the absolute Galois group, and what they "mean". An answer was given many years ago by Ax (I guess in one of his Annals papers, 1968 or 1969), and there are elaborations and new results in the thesis of James Gray, now published in the J. of Symbolic Logic as "Coding complete theories in Galois groups" (his thesis is also available freely online).

The reason why these "other conjugacy classes" (which are essentially all of them) are difficult to construct is that the associated fixed fields are the algebraic (over $Q$) subfields of pseudofinite fields of characteristic zero. In Theorem 1.27 of his thesis, Gray states that there is a [natural, homeomorphic in the appropriate topology] bijection between the set of conjugacy classes in $Gal(bar Q / Q)$ and the Stone space of completions of $ACFA_0$ -- the theory of algebraically closed fields of characteristic zero with generic automorphism. Given such a completion of $ACFA_0$, realized by a model $(K, sigma)$ where $K$ is algebraically closed, and $sigma$ is a generic automorphism (see MacIntyre, "Generic automorphisms of fields" for the definition of generic used here), the fixed field $K^sigma$ is a pseudofinite field of characteristic zero.

Now, while two conjugacy classes in $Gal(bar Q / Q)$ are quite simple to describe -- the trivial conjugacy class and the conjugacy class of order 2 elements -- other conjugacy classes contain elements of infinite order. The associated fixed fields (in this infinite-order case) in $bar Q$ are psuedofinite fields, which are difficult to get your hands on. Probably the best way is to take an ultraproduct (for a non-principal ultrafilter on the set of prime numbers) of finite fields, and take the algebraic elements within. This is certainly nonconstructive, relying heavily on Zorn's lemma.

Still, such fields have arithmetic significance. The model theory related to $ACFA_0$ has had a great deal of impact on number theory lately, and Fried-Jarden also touch on related matters in their "Field Arithmetic" book.