素数与不可约多项式
Primes and irreducible polynomials
AI总结：
本文在Murty关于素数k进制展开对应多项式不可约的定理基础上，证明乘数t小于k时结论仍成立，并给出控制多项式因子的技巧，旨在拓展这类多项式的相关结论。
摘要：
2002年，Murty证明：若素数p的k进制展开为p=∑（i=0到n）a_i k^i，则多项式f(x)=∑（i=0到n）a_i x^i在ℤ[x]中是不可约的；当k=10时，该结论是某文献的结果。本文作者认为这类多项式很有趣，值得进一步研究，计划挖掘更多相关结论。文章第一节证明了该定理的更强版本：若将素数p乘以一个小于k的因子t，结论仍成立；第二节进一步考虑更大的乘数t，给出了控制多项式某一因子的技巧。
英文摘要：
In 2002, M.Ram Murty showed that if p is a prime with k-adic expansion :$p = \sum_{i = 0}^n a_i k^i$ , then the polynomial $f(x) = \sum_{i = 0}^n a_ix^i$ is irreducible in $\mathbb{Z}[x]$.When $k = 10$ , it's a result of A.Cohn. I think this kind of polynomials is really interesting and worse to speak more. So I plan to find more conclusions about this kind of polynomials. In the first section of this article, author proves a stronger version of this theorem that if we multiply prime $p$ by a factor $t$ that is smaller than $k$ ,the conclusion also holds. In the second section, author further consider larger multiplier $t$ ,and gives a technique to control one of the factors of the polynomial.
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