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period lattice generator of thetensor powers of the Carlitz module which was already observed by M.Papanikolas in a yet unpublished paper.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

Algebraic independence of the Carlitz period and its hyperderivatives