论文标题
指数泰勒多项式总和的代数特性
Algebraic properties of summation of exponential Taylor polynomials
论文作者
论文摘要
令$ n \ ge 1 $为整数，$ e_n（x）$表示截短的指数taylor polyenmial，即$ e_ {n}（x）= \ sum_ {i = 0}^n \ frac {x^i} {x^i} {i！} $。 Schur的著名定理指出，如果$ \ Q $ $ e_n（x）$的GALOIS组是交替的组$ a_n $，如果$ n $由4分组或对称组$ s_n $否则。在本文中，我们研究了两个截断的指数taylor polyenmials $ \ e_n（x）的总和的代数属性：= 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 $ 2 $但$ | C_0 | $的功率为2的正力。这扩展了Schur的另一个定理。我们还表明，如果$ n \ not \ in \ {2,4 \} $，$ \ e_n（x）$是不可约的。此外，我们表明$ {\ rm gal} _ {\ q}（\ e_n）$包含$ a_ {n} $，但$ n = 4 $，在这种情况下，$ {\ rm gal} _ {\ q}（\ q}（\ e_4）= s_3 $。最后，我们证明了Galois组$ {\ rm gal} _ {\ q}（\ e_n）$是$ s_n $，如果$ n \ equiv 3 \ equiv 3 \ pmod 4 $，或者$ n $均匀且$ v_p（n！n！）$ of $ n-1 $ and o e equal n o e equal and equal and equal and pmod and pmod 4 $ p $，与$ \ sum_ {i = 1}^{p-1} 2^{p-1-i} i！$和正整数到$ p $。
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 $p$.
