In this article we show that the coordinates of a period lattice generator of the n-th tensor power of the Carlitz module are algebraically independent, if n is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for t-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another ingredient is a theorem which shows hypertranscendence for the Anderson-Thakur function ω(t), i.e.

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... hat ω(t) and all its hyperderivatives with respect to t are algebraically independent.