Geometric Galois theory, nonlinear number fields and a Galois group interpretation of the idele class group

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over Q, a hyperbolized adele class group G(K) is assigned to every number field K/Q. The projectivization of the Hardy space PH.[K] of graded-holomorphic functions on G(K) possesses two operations circle plus and circle times giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that Gal(PH.[K]/K) = 1 and Gal(PH.[L]/PH.[K]) congruent to Gal(L/K) if L/K is Galois. If K-ab denotes the maximal abelian extension of K and C-K is the idele class group, it is shown that there are embeddings of C-K into Gal circle plus(PH.[K-ab]/K) and Gal circle times(PH.[K-ab]/K), the "Galois groups" of automorphisms preserving circle plus (respectively, circle times) only.