Arithmetic actions on cyclotomic function fields

TL;DR

This paper investigates the structure of cyclotomic function fields generated via Carlitz actions, analyzing Galois groups and invariants, revealing that their invariant rings are not polynomial.

Contribution

It provides a detailed group-theoretic analysis of cyclotomic function fields and characterizes the nature of their invariant rings, highlighting new structural insights.

Findings

Galois group structures for cyclotomic function fields are explicitly derived.

Tame and wild ramification structures are distinguished and described.

Invariant rings associated with these fields are shown to be non-polynomial.

Abstract

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.