Geometric Galois Theory, Nonlinear Number Fields and a Galois Group Interpretation of the Idele Class Group

Abstract

This paper concerns the description of holomorphic extensions of algebraic number fields. We define a hyperbolized adele class group for every number field K Galois over Q and consider the Hardy space H[K] of graded-holomorphic functions on the hyperbolized adele class group. We show that the hyperplane N[K] in the projectivization PH[K] defined by the functions of non-zero trace possesses two partially-defined operations + and x, with respect to which there is canonical monomorphism of K into N[K]. We call N[K] a nonlinear field extension of K. We define Galois groups for nonlinear fields and show that Gal(N[L]/N[K]) is isomorphic to Gal(L/K) if L/K is Galois. If Q^{ab} denotes the maximal abelian extension of Q, C(Q) the idele class group and $\bar{N}[Q^{ab}]=PH[K] is the full projectivization, then there are embeddings of C(Q) into Gal_{+}(\bar{N}[Q^{ab}]/Q) and Gal_{x}(\bar{N}[Q^{ab}]/Q), the "Galois groups" of automorphisms preserving + resp. x only.