# Irreducibility criterion for certain trinomials

## TL;DR

This paper investigates the irreducibility of specific trinomials, providing new criteria and correcting previous results, with a focus on polynomials of the form x^n + ε_1 x^m + ε_2.

## Contribution

It introduces an irreducibility criterion for certain trinomials and corrects earlier findings by W. Ljuggren regarding their reducibility and cyclotomic factors.

## Key findings

- Proves irreducibility for x^n + ε_1 x^m + p^k ε_2 when m=1.
- Provides cyclotomic factorization and reducibility criteria for x^n + ε_1 x^m + ε_2.
- Corrects previous results on the reducibility of these trinomials.

## Abstract

In this article we study the irreducibility of polynomials of the form $x^n+\epsilon_1 x^m+p^k\epsilon_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the form $x^n+\epsilon_1x^m+\epsilon_2$, where $\epsilon_i\in \{\, -1,+1\,\}$. This corrects few of the existing results of W. Ljuggren's on $x^n+\epsilon_1x^m+\epsilon_2$.