Function Fields of Class Number One

TL;DR

This paper completes the classification of function fields over finite fields with positive genus and class number one, confirming that only the previously known seven (plus one additional) such fields exist.

Contribution

The paper corrects and finalizes the list of function fields with class number one, proving that the 8th example found by Strirpe was already obtainable by previous methods.

Findings

Confirmed the list of function fields with class number one is complete.

Identified the 8th example as previously obtainable by existing methods.

Provided a corrected proof for the classification.

Abstract

In 1975, [LMQ] listed 7 function felds over fnite felds (up to isomorphism) with positive genus and class number (i.e., the size of the divisor class group of degree zero) one and claimed to prove that these were the only ones such. In [S1], Claude Strirpe found 8th one! In this paper, we fix the argument in [LMQ] to show that this 8th example could have been found by [LMQ] method and is the only one, so that the list is now complete.