Perturbing Eisenstein polynomials over local fields

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension. Let $\pi_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $\pi_L$ over $K$. Suppose $\tilde{\pi}_L$ is another uniformizer for $L$ such that $\tilde{\pi}_L\equiv\pi_L+r\pi_L^{\ell+1} \pmod{\pi_L^{\ell+2}}$ for some $\ell\ge1$ and $r\in O_K$. Let $\tilde{f}(X)$ be the minimum polynomial for $\tilde{\pi}_L$ over $K$. In this paper we give congruences for the coefficients of $\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. These congruences improve and extend work of Krasner.