# Defectless polynomials over henselian fields and inductive valuations

## TL;DR

This paper establishes a canonical bijection between a discrete MacLane space and the set of defectless monic irreducible polynomials over a henselian valued field, using inductive valuations to classify these polynomials.

## Contribution

It introduces a new correspondence linking defectless polynomials to inductive valuations via a canonical bijection, advancing the understanding of polynomial factorization over henselian fields.

## Key findings

- Canonical bijection between MacLane space and defectless polynomials
- Classification of defectless polynomials via inductive valuations
- Enhanced understanding of polynomial structure over henselian fields

## Abstract

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on $\,\mathbb{P}^{dless}$, we establish a canonical bijection $\mathbb{M}\to \mathbb{P}^{dless}/\!\!\approx$, where $\mathbb{M}$ is a discrete MacLane space, constructed in terms of inductive valuations on $K[x]$ extending $v$.