A long-established theorem of Schur states that the polynomial \(\sum\limits_{i=0}^{n} a_i\frac{x^i}{i!}\) is irreducible over the field \(\mathbb{Q}\) of rational numbers for all \(n \geq 1\) when each \(a_i \in \mathbb{Z}\) and \(|a_0| = |a_n| = 1\). An alternate proof of this result when \(|a_i|=1\) for \(i\in\{0,1,\dots,n\}\) was given by Coleman in 1987. In this lecture, after introducing Newton polygons and \(\phi\)-Newton polygons, we shall discuss some applications of these to obtain generalisations of the well known Eisenstein-Dumas Irreducibility Criterion. We shall also discuss recently proved extended versions of the above mentioned results of Schur and Coleman.
