Let be an extension of the -adic numbers with uniformizer . Let and be Eisenstein polynomials over of degree that generate isomorphic extensions. We show that if the cardinality of the residue class field of divides , then . This makes the first (nonzero) digit of the -adic expansion of an invariant of the extension generated by . Furthermore we find that noncyclic extensions of degree of the field of -adic numbers are uniquely determined by this invariant.
"The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of $p$-adic fields." Involve 13 (5) 747 - 758, 2020. https://doi.org/10.2140/involve.2020.13.747