2024/06/27 by Alexander I. Efimov, Efimov, Alexander I.
Mathematics · #11S31 #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2406.19163
openalex publication_date 2024/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the field of Hahn-Witt series HW(\mathbbFp) with residue field \mathbbFp (also known as a p-adic Malcev-Neumann field \citeLa86, P93), and its generalizations. Informally, the Hahn-Witt series are possibly infinite linear combinations of rational powers of p, in which the coefficients are Teichmüller 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 \mathbbFp. 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)(\mathbbFq), where π is a uniformizer in a local field K with residue field \mathbbFq. Again, this field is an algebraically closed extension of K, and it has a canonical automorphism φπ, coming from the relative Frobenius of \mathbbFq over \mathbbFq. We prove that the action of φπ on the maximal abelian extension Kab corresponds via local class field theory to the uniformizer -π∈ K^*.