On enumerating extensions of p-adic fields with given invariants

Abstract

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.

Computing the Galois group of a polynomial over a $p$-adic field

“…Recall (e.g. [19] or [8]) that to an extension of p-adic fields, we can attach a ramification polygon, which is an invariant of the extension. By attaching further residual information such as the residual polynomials of each face of the ramification polygon, we can form a finer invariant.…”

“…Using the pAdicExtensions package [9], which implements these invariants, we generated all possible equivalence classes of the finest such invariant, called the fine ramification polygon with residues and uniformizer residue in [8], for totally ramified extensions of degree 16 of Q 2 .…”