On nth order Euler polynomials of degree n that are Eisenstein

Abstract

For $m$ an even positive integer and $p$ an odd prime, we show that the generalized Euler polynomial $E_{mp}^{\left(mp\right)}\left(x\right)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m(2^m-1)B_m$. As a consequence, we deduce that at least $1/3$ of the generalized Euler polynomials $E_n^{\left(n\right)}\left(x\right)$ are in Eisenstein form with respect to a prime $p$ dividing $n$ and, hence, irreducible over $Q$.