Sudesh Kaur Khanduja
Indian Institute of Science Education and Research, Mohali; and Panjab University, Chandigarh
Title
Newton Polygons and Irreducibility of Polynomials with Integer Coefficients
Abstract
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.
Zoom Link: https://osu.zoom.us/j/94316892771?pwd=uyeg0zJ8zDU2S2hMEV0EayzKq8t8Ao.1