On the field generated by the periods of a Drinfeld module

Generalizing the results of Maurischat in \cite{Maurischatxx}, we show that the field $K_{\infty}(\Lambda)$ of periods of a Drinfeld module $\phi$ of rank $r$ defined over $K_{\infty} = \mathds{F}_{q}((T^{-1}))$ may be arbitrarily large over $K_{\infty}$. 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 $r$.