On the Factorisation of Polynomials

Abstract

The division algorithm for polynomials provides a notion of divisibility in $\mathbb K[X]$ when $\mathbb K=\mathbb Q$,$\mathbb R$ ou $\mathbb C$, and such a notion enjoys properties analogous to those of the corresponding concept in $\mathbb Z$. It is then natural to ask whether there exists some notion of primality in $\mathbb K[X]$, which furnishes some sort of unique factorisation with properties similar to the unique factorisation of integers. Our purpose in this chapter is to give precise answers to these questions, which shall encompass polynomials with coefficients in $\mathbb Z_p$, for some prime integer p.