Polynomial factorization over henselian fields

TL;DR

This paper surveys various mathematical ideas and algorithms related to polynomial factorization over henselian fields, focusing on the development of methods to factor irreducible polynomials in such valued fields.

Contribution

It compiles and reviews the key concepts and algorithms from multiple researchers, providing a comprehensive overview of polynomial factorization over henselian fields.

Findings

Unified framework for polynomial factorization algorithms

Conditions under which the algorithms succeed

Connections between different mathematical approaches

Abstract

Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaqui\'e and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.