Finiteness of fixed-degree extensions of a p-adic field

A p-adic field has only finitely many extensions of a fixed degree up to isomorphism. There are finitely many possible residue degrees and a unique unramified extension of each such degree. Over each maximal unramified subfield, a totally ramified extension is generated by a root of an Eisenstein polynomial. The coefficient space of these polynomials of fixed degree is compact. Hensel lemma and Krasner's lemma show that sufficiently close polynomials generate isomorphic extensions, giving a finite cover by neighborhoods of a constant extension type.