Enumeration of Isomorphism Classes of Extensions of p-adic Fields

الأصل · EN

Let Ω be an algebraic closure of Qₚ and let F be a finite extension of Qₚ contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number I(F,f,e) of F-isomorphism classes of such extensions. We determine I(F,f,e) completely when p² e and get partial results when p²∥ e. When s is large, I(Qₚ,f,e) is equal to the number of isomorphism classes of finite commutative chain rings with residue field Fₚᶠ, ramification index e, and length s.