On Towers and Composita of Towers of Function Fields over Finite Fields

For a towerF1 F2 ••• of algebraic function fieldsFi/ q, define λ = limi→∞N(Fi)/g(Fi), whereN(Fi) is the number of rational places andg(Fi) is the genus ofFi/ q. The tower is said to be asymptotically good if λ > 0. We give a very simple explicit example of an asymptotically good tower for all non-prime fields q. In this example, all stepsFi+1/Fiare tamely ramified Kummer extensions. We then show that any function fieldF/ qhaving at least one rational place can be embedded into an asymptotically good tower, and we study the behaviour of λ in the compositum of a towerF1 F2 ••• with an extensionE/F1.