A combinatorial proof of the Kronecker--Weber Theorem in positive characteristic

TL;DR

This paper provides a combinatorial proof of the Kronecker--Weber Theorem for global fields of positive characteristic, utilizing Witt vectors and counting arguments to compare extension counts with cyclotomic function fields.

Contribution

It introduces a new combinatorial proof of the theorem in positive characteristic using Witt vectors and counting methods, differing from previous algebraic approaches.

Findings

Count of p-cyclic extensions with fixed degree and bounded conductor matches the count of subextensions in cyclotomic function fields.

The proof confirms the correspondence between certain field extensions and cyclotomic subextensions in positive characteristic.

Utilizes Witt vectors to develop arithmetic tools for counting and comparing field extensions.

Abstract

In this paper we present a combinatorial proof of the Kronecker--Weber Theorem for global fields of positive characteristic. The main tools are the use of Witt vectors and their arithmetic developed by H. L. Schmid. The key result is to obtain, using counting arguments, how many $p$--cyclic extensions exist of fixed degree and bounded conductor where only one prime ramifies are there. We then compare this number with the number of subextensions of cyclotomic function fields of the same type and verify that these two numbers are the same.