Elementary abelian \(p\)-extensions of algebraic function fields (Q1814168)

Let \(K\) be a field of characteristic \(p >0\), and let \(F/K\) be an algebraic function field. If \(E/F\) is an elementary abelian Galois extension with Galois group of order \(p^ n\) then \(E\) can be generated over \(F\) by an element \(y\) whose minimal polynomial is of the form \(T^{p^ n}-T-y\). Furthermore a formula for the genus of \(E\) is derived, from which it can be seen that the genus of \(E\) grows faster then the number of rational points when the degree of \(E\) over \(F\) goes to infinity (from the viewpoint of coding theory this is a disappointing result). The authors end with giving a new example of a function field \(E/K\) with non-classical gap number.