Class number one problem for imaginary function fields: the cyclic prime power case
Let \(\mathbb F_q\) be a finite field having \(q\) elements. In this paper, the author finds all (up to \(\mathbb F_q\)-isomorphisms) imaginary cyclic extensions of prime power degree \(K/\mathbb F_q(T)\) such that the integral closure of \(\mathbb F_q[T]\) in \(K\) is a principal ideal domain. This extends the results of \textit{R. E. MacRae} [J. Algebra 17, 243--261 (1971; Zbl 0212.53302)]. In order to do this, the author uses the results of \textit{G. Lachaud} and \textit{M. Martin-Deschamps} [Acta Arith. 56. 329--340 (1990; Zbl 0727.14019)].
imaginary cyclic extensions of prime power degree
scientific article; zbMATH DE number 1533163