# Cubic Fields: A Primer

## TL;DR

This paper provides a comprehensive classification of cubic field extensions, including Galois cases, and analyzes ramification and polynomial decomposition over finite fields, using explicit forms and Riemann-Hurwitz formulas.

## Contribution

It introduces an explicit classification method for all cubic extensions and details ramification behavior in global function fields.

## Key findings

- Complete classification of cubic extensions via three fundamental forms
- Explicit determination of ramification and splitting in cubic extensions
- Decomposition of cubic polynomials over finite fields

## Abstract

We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a field. The splitting and ramification of places in a separable cubic extension of any global function field are completely determined, and precise Riemann-Hurwitz formulae are given. In doing so, we determine the decomposition of any cubic polynomial over a finite field.