Abstract

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 [Formula: see text] is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧•[K] of graded-holomorphic functions on [Formula: see text] possesses two operations ⊕ and ⊗ 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]) ≅ 𝖦𝖺𝗅(L/K) if L/K is Galois. If K ab denotes the maximal abelian extension of K and 𝖢K is the idele class group, it is shown that there are embeddings of 𝖢K into 𝖦𝖺𝗅⊕(ℙ𝖧•[K ab ]/K) and 𝖦𝖺𝗅⊗(ℙ𝖧•[K ab ]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

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