Question

Let $k$ be a finite field of characteristic $p$. There is only one characteristic-$p$ local field with residue field $k$, namely $k((\pi))$, where $\pi$ is transcendental, and there is a "smallest" characteristic-$0$ local field with residue field $k$, namely the degree-$a$ unramified extension $K$ of $\mathbb{Q}_p$, where $q=p^a$ is the cardinality of $k$.

Every characteristic-$0$ local field with residue field $k$ is a finite totally ramified extension of $K$, and every finite totally ramified extension of $K$ has residue field $k$.

As $K$ has only finitely many extensions of any given degree, we might ask for the number of totally ramified extensions $L|K$ of given degree $[L:K]=n$. Serre's "mass formula" (Comptes rendus 1978), says that there are exactly $n$ such extensions, when counted properly. This formula was also proved by Krasner by his methods.

Let Serre explain:

"Let $K$ be a local field, with finite residue field with $q$ elements. Let $n$ be a positive integer, and let $\Sigma_n$ be the set of all totally ramified extensions of $K$ of degree $n$ contained in a given separable closure of $K$. If $(n,p)=1$, \operatorname{gcd}(n,p)=1$, it is easy to show that $\text{Card}(\Sigma_n)=n$. If $L$ belongs to $\Sigma_n$, put $c(L)=d(L)-n+1$, where $d(L)$ is the valuation of the discriminant of $L/K$. Our 'mass formula' is

$\sum_{L\in \Sigma_n} q^{-c(L)}=n$.

We give two proofs. The first one uses Eisenstein polynomials, while the second one applies the H. Weyl integration formula to the multiplicative group of a division algebra." MR0500361 (80a:12018)

(Note that $k((\pi))$ has many totally ramified extensions $L$, but they are all of the form $L=k((\varpi))$ for some uniformiser $\varpi$ of $L$.)