Pólya and Newtonian function fields (Q930583)

Let \(K/\mathbb{F}_q(T)\) be a finite extension of the field of rational functions over a finite field \(\mathbb{F}_q\), and let \(O_K\) be the integral closure of \(\mathbb{F}_q[T]\) in \(K\). Moreover let \(Int(O_K)\) denotes the \(O_K\)-module of \(K\)-polynomials \(f\) with \(f(O_K)\subset O_K\). The field \(K\) is called a Pólya field, if \(\text{Int}(O_K)\) has a basis \(f_0,f_1,\dots\) with \(\deg f_n=n\) for every \(n\). The author characterizes Kummerian and Artin-Schreier-Witt extensions with this property and shows that every cyclotomic extension has it as well.