Computations in Cubic Function Fields of Characteristic Three

TL;DR

This paper studies cubic function fields over characteristic three, establishing a standard form, computing invariants like discriminant and genus, and developing algorithms for ideal class group arithmetic.

Contribution

It introduces a standard form for cubic curves in characteristic three and provides algorithms for arithmetic in their ideal class groups.

Findings

Computed field discriminants and genus for these fields

Described splitting behavior of places

Developed composition and reduction algorithms

Abstract

This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of these function fields, we are able to calculate the field discriminant and the genus. We also describe the splitting behavior of any place, and give composition and reduction algorithms for arithmetic in the ideal class group.