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

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(f)− disc(g)) > v(discf). This makes the first (non-zero) digit of the π-adic expansion of disc(f) an invariant of the extension generated by φ. Furthermore we find that non-cyclic extensions of degree p of the field of p-adic numbers are uniquely determined by this invariant.