sanctany 发布于四月 4, 2022 分享 发布于四月 4, 2022 · 只看该作者 这是我最近在研究的一个课题， 关于对任意单调多项式 $X^n + a_{1}X^{n-1} + \dots + a_{n-2}X^{1} + a_{n-1}$, where $x \in \mathbb{Z}$ $a_{1} \dots a_{n-1} \in \mathbb{Z}$ 根据拟定n和每项系数的取值范围，我们可得有限数量的多项式，其中部分为不可分多项式(irreducible)，根据Galois群的定义，我们易得其存在Galois群同构于某些有限单群。 例如当 n =3, 各项系数取值为[2,-2], 那么总共存在125个多项式，其中72个为不可分多项式，72个里有68个同构于S3，4个同构于C3。 关于这个我写了个程序导出对应取值的多项式，输出如下： 对于n=2, 系数取值[-10,10] 引用 We have in total 9 polynomials We have 5 irreducible polynomials ('C2', 5) We have in total 25 polynomials We have 15 irreducible polynomials ('C2', 15) We have in total 49 polynomials We have 33 irreducible polynomials ('C2', 33) We have in total 81 polynomials We have 56 irreducible polynomials ('C2', 56) We have in total 121 polynomials We have 90 irreducible polynomials ('C2', 90) We have in total 169 polynomials We have 128 irreducible polynomials ('C2', 128) We have in total 225 polynomials We have 178 irreducible polynomials ('C2', 178) We have in total 289 polynomials We have 232 irreducible polynomials ('C2', 232) We have in total 361 polynomials We have 295 irreducible polynomials ('C2', 295) We have in total 441 polynomials We have 365 irreducible polynomials ('C2', 365) 对于n=3，系数取值[-10,10]: 引用 We have in total 27 polynomials We have 12 irreducible polynomials ('S3', 12) We have in total 125 polynomials We have 72 irreducible polynomials ('S3', 68) ('C3', 4) We have in total 343 polynomials We have 226 irreducible polynomials ('S3', 216) ('C3', 10) We have in total 729 polynomials We have 514 irreducible polynomials ('S3', 496) ('C3', 18) We have in total 1331 polynomials We have 1002 irreducible polynomials ('S3', 976) ('C3', 26) We have in total 2197 polynomials We have 1704 irreducible polynomials ('S3', 1668) ('C3', 36) We have in total 3375 polynomials We have 2718 irreducible polynomials ('S3', 2670) ('C3', 48) We have in total 4913 polynomials We have 4036 irreducible polynomials ('S3', 3972) ('C3', 64) We have in total 6859 polynomials We have 5756 irreducible polynomials ('S3', 5654) ('C3', 102) We have in total 9261 polynomials We have 7878 irreducible polynomials ('S3', 7760) ('C3', 118) 对于n=4的情况我的电脑不允许我得出数据（系数取值超过3之后需要半个多小时才能算出结果）. 我之前以为最简单的关于多项式数量的规律，是p^n， p为素数且大于等于3。但是在n=3的里面出现了9不是素数的情况。 其他的我完全看不出来有啥规律，求大佬们一起帮忙看看！ 非常感谢！ 链接到点评
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