The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of \(p\)-adic fields
This article takes interest to the classification of extensions of \(p\)-adic fields using invariants. The authors introduce a new invariant and show how it is related to other invariants of the extension. Commonly, only the valuation of the discriminant is used as an invariant of extensions of a \(p\)-adic field, since it yields information about the ramified part of the extension. It is natural to ask whether this invariant can be refined by considering not the complete discriminant but possibly some of the digits of its \(p\)-adic expansion. To this end the authors restrict their investigation to discriminants of integral bases of a certain form. They find that the first digit of the \(p\)-adic expansion of the discriminant of generating Eisenstein polynomials is an invariant for most extensions. This allows for a finer classification of totally ramified extensions. The structure of the article is organized in two parts, the first section presents the main results of the paper and the second contains proofs to those results.
scientific article; zbMATH DE number 7364869