On the Hahn-Witt series and their generalizations

In this paper we study the field of Hahn-Witt series HW (Fₚ) with residue field Fₚ (also known as a p-adic Malcev-Neumann field La86, P93), and its generalizations. Informally, the Hahn-Witt series are possibly infinite linear combinations of rational powers of p, in which the coefficients are Teichm\"uller representatives, and the set of exponents is well-ordered. They form an algebraically closed extension of Qₚ, with a canonical automorphism, coming from the absolute Frobenius of Fₚ. We prove that the action of on the p-power roots of unity is given by () =^-1, answering a question of Kontsevich. More generally, we consider the -typical Hahn-Witt series HW (₊, ) (Fq), where is a uniformizer in a local field K with residue field Fq. Again, this field is an algebraically closed extension of K, and it has a canonical automorphism _, coming from the relative Frobenius of Fq over Fq. We prove that the action of _ on the maximal abelian extension K^ab corresponds via local class field theory to the uniformizer - K^*.