$$1 + \frac{1}{{{T^2} - T}} + \frac{1}{{({T^4} - T){{({T^2} - T)}^2}}} + \frac{1}{{({T^8} - T){{({T^4} - T)}^2}{{({T^2} - T)}^4}}} + \cdots $$

Abstract

We conclude our tour of classical transcendental number theory by transcending the world of numbers themselves and ascending to the realm of formal power series. Specifically, we consider transcendence issues within the setting of function fields in a single variable over a finite field. While this theory has important implications in many different areas of mathematics, our goal here is to discover an object in this context that is analogous to the all-important exponential function e z .