Algebraic properties of summation of exponential Taylor polynomials

Abstract

Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=\sum_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$ with all $c_i \ (0\le i\le n-1)$ being integers is irreducible over $\Q$ if either $c_0=\pm 1$, or $n$ is not a positive power of $2$ but $|c_0|$ is a positive power of 2. This extends another theorem of Schur. We show also that $\E_n(x)$ is irreducible if $n\not\in\{2,4\}$. Furthermore, we show that ${\rm Gal}_{\Q}(\E_n)$ contains $A_{n}$ except for $n=4$, in which case, ${\rm Gal}_{\Q}(\E_4)=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_n)$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and $v_p(n!)$ is odd for a prime divisor of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals the product of an odd prime number $p$ which is coprime to $\sum_{i=1}^{p-1}2^{p-1-i}i!$ and a positive integer coprime to $p$.