Genus fields of abelian extensions of rational congruence function fields. II
This paper extends and improves on Part I of this paper [Finite Fields Appl. 20, 40--54 (2013; Zbl 1285.11140); corrigendum ibid. 33, 283--285 (2015; Zbl 1368.11120)] to ``compute'' the genus field of an abelian extension of rational congruence function fields. The paper's well-written introduction explains the history and evolution in terms of class field theory. Let $K$ be a global field (i.e., either a finite extension of $\mathbb{Q}$ or a finite extension of $\mathbb{F}(T)$, where $\mathbb{F}$ is a finite field and $T$ is an indeterminate). Whereas the definition of the genus field of $K$ in the number field case has been known since the work of Fröhlich in the 1950s, the function field case is a more recent development. In case $K/\mathbb{Q}$ is a finite extension, the genus field $K_g$ of $K$ is defined to be the composite field $Kk^*$, where $K^*/\mathbb{Q}$ is a maximal abelian number field such that $Kk^*/K$ is unramified. The definition in the function field case is similar but more technical. The main result, Theorem 6.9 which computes $K_g$ as a composite of simpler field extensions, is also too technical to be presented in a short review.
scientific article; zbMATH DE number 6999256
scientific article