2020/11/12 by Lingfeng Ao, Ao, Lingfeng, Shaofang Hong +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2011.06273
openalex publication_date 2020/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n≥ 1 be an integer and en(x) denote the truncated exponential Taylor polynomial, i.e. en(x)=∑i=0n(xi)/(i!). A well-known theorem of Schur states that the Galois group of en(x) over \Q is the alternating group An if n is divisible by 4 or the symmetric group Sn otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials \En(x):=en(x)+en-1(x). We show that (xn)/(n!)+∑i=0n-1ci(xi)/(i!) with all ci (0≤ i≤ n-1) being integers is irreducible over \Q if either c0=± 1, or n is not a positive power of 2 but |c0| is a positive power of 2. This extends another theorem of Schur. We show also that \En(x) is irreducible if n\not∈\2,4\. Furthermore, we show that \rm Gal\Q(\En) contains An except for n=4, in which case, \rm Gal\Q(\E4)=S3. Finally, we show that the Galois group \rm Gal\Q(\En) is Sn if n≡ 3 \pmod 4, or if n is even and vp(n!) is odd for a prime divisor of n-1, or if n≡ 1\pmod 4 and n-2 equals the product of an odd prime number p which is coprime to ∑i=1p-12p-1-ii! and a positive integer coprime to p.