  Let $K$ be a global function field and fix a place $\infty$ of $K$. Let $L/K$
be a finite real abelian extension, i.e. a finite, abelian extension such that
$\infty$ splits completely in $L$. Then we define a group of elliptic units
$C_L$ in $\mathcal{O}_L^\times$ analogously to Sinnott's cyclotomic units and
compute the index $[\mathcal{O}_L^\times:C_L]$. In the second part of this
article, we additionally assume that $L$ is a cyclic extension of prime power
degree. Then we can use the methods from Greither and Ku\v{c}era to take
certain roots of these elliptic units and prove a result on the annihilation of
the $p$-part of the class group of $L$.
