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

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