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

Abstract

We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of ${\mathbb{Q}}_2$ of degrees 18, 20 and 22, tables of which are available online.

39 pages; tables at https://cjdoris.github.io/pAdicGaloisGroupTables; accepted by IJNT