Divisor class number one problem for Abelian extensions over rational function fields
The characterization of cyclotomic function fields and their maximal real subfields with divisor class number one was done by \textit{M. Kida} and \textit{N. Murabayashi} [Tokyo J. Math. 14, 45--56 (1991; Zbl 0734.11060)]. Also \textit{S. Bae} and \textit{P. Kang} [Acta Arith. 102, 251--259 (2002; Zbl 0989.11064)] determined all cyclotomic function fields with relative divisor class number one. This paper determines all abelian extensions of rational function fields with divisor class number one and all imaginary abelian extensions with relative divisor class number one. Let \(k=\mathbb{F}_q(t)\) be the rational function field over the finite field \(\mathbb{F}_q\) with \(q\geq 3\) elements. The author relies on Proposition \(1\) and \(2\) in [Tokyo J. Math. 14, 45--56 (1991; Zbl 0734.11060)] to reduce the problem to finding abelian extensions over \(k\) with genus \(0\) or \(1\). The computation of the genus of a finite abelian extension \(K\) of \(k\) makes use of the Riemann-Hurwitz formula and the relation between the different of \(K/k\) and the ramification groups of \(K/k\). First the author computes all imaginary abelian extensions \(K/k\) with divisor class number one in the cases the genus \(g_K\) of \(K\) is zero and one respectively. Then the author computes all real abelian extensions \(K/k\) with genus zero.
scientific article; zbMATH DE number 5144376