Extensions of local fields given by 3-term Eisenstein polynomials

Let $K$ be a local field with residue characteristic $p$ and let $L\mathrm{/}K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L\mathrm{/}K$ has only two distinct indices of inseparability then there exists a uniformizer ${\pi }_L$ for $L$ whose minimum polynomial over $K$ has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case $k=1$.