  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.
