# An explicit triangular integral basis for any separable cubic extension of a function field

## TL;DR

This paper provides an explicit construction of a triangular integral basis for all separable cubic extensions of rational function fields over finite fields, along with a formula for their discriminants.

## Contribution

It introduces a universal explicit basis for separable cubic extensions of function fields and derives a discriminant formula in a standard Galois closure form.

## Key findings

- Explicit triangular integral basis for all such extensions
- Discriminant formula in standard Galois tower form
- Applicable in any characteristic

## Abstract

We determine an explicit triangular integral basis for any separable cubic extension of a rational function field over a finite field in any characteristic. We obtain a formula for the discriminant of every such extension in terms of a standard form in a tower for the Galois closure.