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 $K$ 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$.) 6 added 147 characters in body Let$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$(n,p)=1$, 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$L$.) 5 edited body 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$a$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$(n,p)=1$, 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) 4 Changed$k[[\pi]]$to$k((\pi))$. 3 adjusted texing as requested by C.S. Dalawat   2 Changed$k[[\pi]]$to$k((\pi))\$.