# Primes and irreducible polynomials

## TL;DR

This paper explores the irreducibility of polynomials derived from prime numbers' k-adic expansions, extending Murty's theorem by considering scaled primes and analyzing polynomial factors.

## Contribution

It proves a stronger version of Murty's theorem for scaled primes and introduces techniques to analyze polynomial factors for larger multipliers.

## Key findings

- Proved irreducibility for scaled primes with multiplier less than k
- Developed methods to control polynomial factors with larger multipliers
- Extended existing theorems on prime-based polynomials

## Abstract

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.