On the automorphisms of the spectral completion of the algebraic numbers field

Abstract

Let G = Gal ( Q ¯ / Q ) be the absolute Galois group of Q and let A = C ( G , C ) be the Banach algebra of all continuous functions defined on G with values in C . Let e ¯ be the conjugation automorphism of C and let B be the R -Banach subalgebra of A consisting of continuous functions f such that f ( e ¯ σ ) = e ¯ f ( σ ) for all σ ∈ G . Let ‖ x ‖ = sup { | σ ( x ) | : σ ∈ G } be the spectral norm on Q ¯ and let Q ˜ be the spectral completion of Q ¯ . Using a canonical isometry between Q ˜ and B we study the structure of the group of R -algebras automorphisms of Q ˜ and the structure of its subgroup Alg ( Q ˜ ) of all automorphisms of Q ˜ which when restricted to Q ¯ give rise to elements of G . We introduce a topology on Alg ( Q ˜ ) and prove that this last one is homeomorphic and group isomorphic to G .