  This paper deals with the fundamental period $\tilde{\pi}$ of the Carlitz
module. The main theorem states that the Carlitz period and all its
hyperderivatives are algebraically independent over the base field
$\mathbb{F}_q(\theta)$. Our approach also reveals a connection of these
hyperderivatives with the coordinates of a period lattice generator of the
tensor powers of the Carlitz module which was already observed by M.
Papanikolas in a yet unpublished paper. Namely, these coordinates can be
obtained by explicit polynomial expressions in $\tilde{\pi}$ and its
hyperderivatives. Papanikolas also gave various presentations of these
expressions which we also prove here.
