Representation of units in cyclotomic 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 $\frac{1-{\zeta }_p^e}{1-{\zeta }_p}$ 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.