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 (eσ ) = 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.
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