On the Hahn-Witt series and their generalizations

In this paper we study the field of Hahn-Witt series HW(F̄ₚ) with residue field F̄ₚ (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 φ, coming from the absolute Frobenius of F̄ₚ. We prove that the action of φ on the p-power roots of unity is given by φ(ζ)=ζ⁻¹, answering a question of Kontsevich. More generally, we consider the π-typical Hahn-Witt series HW(K,π)(F̄q), where π 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 φπ, coming from the relative Frobenius of F̄q over Fq. We prove that the action of φπ on the maximal abelian extension Kᵃᵇ corresponds via local class field theory to the uniformizer -π∈ K^*.