On $n^{\mathrm{th}}$ order Euler polynomials of degree $n$ that are Eisenstein

Abstract

For $m$ an even positive integer and $p$ a prime, we show that the generalized Euler polynomial $E_{mp}^{(mp)}(x)$ 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 $E_n^{(n)}(x)$ are in Eisenstein form with respect to a prime $p$ dividing $n$ and, hence, irreducible over $\mathbb{Q}$.