Irreducibility and Primality in Differentiability Classes

Abstract

In this note, we give criteria for the irreducibility of functions in $ C^{m}\left[ 0,1\right] $, where $ m \in \{1,2,3,...\}\cup\{\infty\} \cup \{\omega\} $. We also discuss irreducibility in multivariable differentiability classes. Moreover, we characterize irreducible functions and maximal ideals in $ C^{\infty}\left[ 0,1\right] $. In fact, irreducible and prime smooth functions are the same, and every maximal ideal of $ C^{\infty}\left[ 0,1\right] $ is principal.