Imaginary bicyclic function fields with the real cyclic subfield of class number one
Bull. Korean Math. Soc. 2008 Vol. 45, No. 2, 375-384 Printed June 1, 2008
Hwanyup Jung Chonbuk National University
Abstract : Let $k = \mathbb F_{q}(T)$ and $\mathbb A = \mathbb F_{q}[T]$. Fix a prime divisor $\ell$ of $q-1$. In this paper, we consider a $\ell$-cyclic real function field $k(\sqrt[\ell]{P})$ as a subfield of the imaginary bicyclic function field $K = k(\sqrt[\ell]{P}, \sqrt[\ell]{-Q})$, which is a composite field of $k(\sqrt[\ell]{P})$ with a $\ell$-cyclic totally imaginary function field $k(\sqrt[\ell]{-Q})$ of class number one, and give various conditions for the class number of $k(\sqrt[\ell]{P})$ to be one by using invariants of the relatively cyclic unramified extensions $K/F_{i}$ over $\ell$-cyclic totally imaginary function field $F_{i} = k(\sqrt[\ell]{-P^{i}Q})$ for $1 \le i \le \ell - 1$.
Keywords : imaginary bicyclic function field, real cyclic function field, class number one
