The Hasse-Witt invariant in some towers of function fields over finite fields

In this article we investigate the p-rank of function fields in several good towers. To do this we first recall and establish some properties of the behaviour of the p-rank under extensions. Then we compute the p-ranks of function fields in several optimal towers over a quadratic field $ \mathbb{F}_{q^2 } $ \mathbb{F}_{q^2 } , as well as for a specific good tower over a cubic field $ \mathbb{F}_{q^3 } $ \mathbb{F}_{q^3 } , which was introduced by Bassa, Garcia and Stichtenoth.