Mathematics – Number Theory
Scientific paper
2005-06-24
International Journal of Mathematics, Vol. 16, No. 6 (July 2005)
Mathematics
Number Theory
This is the revised version of an article which appeared by the same name in the International Journal of Mathematics, Vol. 16
Scientific paper
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.
Gendron T. M.
Verjovsky Alberto
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