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

Abstract

Let $K$ be an extension of the $p$-adic numbers with uniformizer $\pi$. Let $\phi$ and $\psi$ 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 $n\left(n-1\right)$, then $v\left(disc\left(\phi \right)-disc\left(\psi \right)\right)>v\left(disc\left(\phi \right)\right)$. This makes the first (nonzero) digit of the $\pi$-adic expansion of $disc\left(\phi \right)$ an invariant of the extension generated by $\phi$. Furthermore we find that noncyclic extensions of degree $p$ of the field of $p$-adic numbers are uniquely determined by this invariant.