The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of p-adic fields

Abstract

Listen

Translate article

Let K be an extension of the p-adic numbers with uniformizer π. Let φ and ψ be Eisenstein polynomials over K of degree n that generate isomorphic extensions. We show that if the cardinality of the residue class field of K divides n(n−1), then v(disc(φ)− disc(ψ))>v(disc(φ)). This makes the first (nonzero) digit of the π-adic expansion of disc(φ) an invariant of the extension generated by φ. Furthermore we find that noncyclic extensions of degree p of the field of p-adic numbers are uniquely determined by this invariant.