Representation of units in cyclotomic function fields

TL;DR

This paper extends Newman's refinement of Hilbert's Satz 90 from cyclotomic number fields to cyclotomic function fields, providing a unique representation of units of norm 1 as quotients of conjugate units.

Contribution

It develops a function field analogue of Newman's result, offering a new criterion for expressing units of norm 1 as quotients of conjugate units in cyclotomic function fields.

Findings

Established a function field analogue of Newman's theorem

Provided a necessary and sufficient condition for units of norm 1

Proved a refinement of Hilbert's Satz 90 in function fields

Abstract

Hilbert's Satz 90 tells us that for a given cyclic extension $K/k$, a unit of norm $1$ in $K$ can be written as a quotient of conjugate elements in $K$. For the extensions $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ with $p$ prime $> 3$, Newman proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{Q}(\zeta_p)$ can be written as a quotient of conjugate units. In order to obtain this result, Newman proved a stronger result that gives a unique representation of units of norm $1$ as a product of a power of $\dfrac{1 - \zeta_p^e}{1 - \zeta_p}$ with a quotient of conjugate units, where $e$ is a given primitive root modulo $p$. In this paper, we obtain a function field analogue of Newman's result for the $\wp$-th cyclotomic function field extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$, where $\wp$ is a monic prime in $\mathbb{F}_q[T]$. As a consequence, we proved a refinement of Hilbert's Satz 90 for the extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$ that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{K}_{\wp}$ can be written as a quotient of conjugate units.