On the computation of all extensions of a \(p\)-adic field of a given degree (Q2723534)

Let \(p\) be a rational prime and let \(\mathbb{Q} _p\) be the field of \(p\)-adic rational numbers. For a finite extension \(k/\mathbb{Q} _p\), the description of the lattice of abelian extensions of \(k\) in an algebraic closure of \(\mathbb{Q} _p\) is given by local class field theory. In the general case, only the formula found by \textit{M. Krasner} [Colloques Int. Centre Nat. Rech. Sci. 143, 143-169 (1966; Zbl 0143.06403)] gives the number of extensions of a given degree. It is possible to adapt Krasner's methods to obtain a set of polynomials defining all of these extensions. This is the aim of this paper. NEWLINENEWLINENEWLINEIn fact, for \(m > 1, d\geq 0\), the authors give an algorithm to compute all extensions of degree \(m\) and discriminant \({\mathfrak p} ^d\) where \({\mathfrak p}\) is the prime ideal of \(k\). The method is reduced to the computation of totally ramified extensions. NEWLINENEWLINENEWLINEThe paper runs as follows. First conditions stated that give all possible discriminants \({\mathfrak p} ^d\) of totally ramified extensions of degree \(m\). Next, the authors introduce an ultrametric distance on the set of Eisenstein polynomials of degree \(m\), which is used in the construction of a set of polynomials defining all totally ramified extensions. Finally, the algorithms of the computation of a minimal set of polynomials generating all the extensions of degree \(m\) and discriminant \({\mathfrak p} ^d\) are given.NEWLINENEWLINENEWLINEThe paper finishes giving some examples and discussing future developments, in particular the algorithm can be refined to the computation of all \(p\)-extensions with a given Galois group using the formulas of \textit{I. R. Shafarevich} [Transl., II. Ser., Am. Math. Soc. 4, 59-72 (1956; Zbl 0071.03302)] and \textit{M. Yamagishi} [Proc. Am. Math. Soc. 123, 2373-2380 (1995; Zbl 0830.11045)].