Classification of algebraic function fields with class number one (Q2343185)

In [J. Number Theory 7, 11--27 (1975; Zbl 0318.12009)], \textit{J. R. C. Leitzel}, \textit{M. L. Madan} and \textit{C. S. Queen} found seven function fields of genus \(g>0\) with class number one over any finite field \(\mathbb F_q\). These had \(g=1,2\) or \(3\), and the authors claimed to have excluded the only other possibility, which has \(g=4\). \textit{M. L. Madan} and \textit{C. S. Queen} [Acta Arith. 20, 423--432 (1972; Zbl 0215.36102)] had previously shown that any other example with \(g>0\) should have \(g=4\) and have exactly one place of degree \(4\) and no places of smaller degree. Such a field was later found by the second author of this paper [J. Number Theory 143, 402--404 (2014; Zbl 1296.11144)], but he could not prove it was unique up to isomorphism. The paper under review fills this gap, and Theorem 1.1 thus gives a complete classification of function fields of genus \(g>0\) with class number one over any finite field \(\mathbb F_q\). The proof of uniqueness uses basic facts about function fields and class field theory and is given in Section 3. The authors also give another proof in Section 4, which is a complete version of the argument of Leitzel, Madan and Queen [loc. cit.]. They also mention that \textit{Q. Shen} and \textit{S. Shi} [J. Number Theory 154, 375--379 (2015; Zbl 1320.11107)] independently proved uniqueness at the same time using a simplified argument.