Enumeration of Isomorphism Classes of Extensions of p-adic Fields

Let $\mathrm{\Omega }$ be an algebraic closure of ${\mathbb{Q}}_p$ and let $F$ be a finite extension of ${\mathbb{Q}}_p$ contained in $\mathrm{\Omega }$. Given positive integers $f$ and $e$, the number of extensions $K\mathrm{/}F$ contained in $\mathrm{\Omega }$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number $\mathfrak{I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine $\mathfrak{I}(F,f,e)$ completely when $p^2\nmid e$ and get partial results when $p^2\parallel e$. When $s$ is large, $\mathfrak{I}({\mathbb{Q}}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\mathbb F}_{p^f}$,ramificationindex$e$,andlength$s$.