On the field generated by the periods of a Drinfeld module

Generalizing the results of Maurischat in \cite{Maurischatxx}, we show that the field $K_{\mathrm{\infty }}(\mathrm{\Lambda })$ of periods of a Drinfeld module $\varphi$ of rank $r$ defined over $K_{\mathrm{\infty }}=\text{mathds}{F}_q((T^{-1}))$ may be arbitrarily large over $K_{\mathrm{\infty }}$. We also show that, in contrast, the residue class degree $f( K_{\infty}(\Lambda) | K_{\infty})$ remains bounded by a constant that depends only on $r$.