= 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.
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