Extended genus field of cyclic Kummer extensions of rational function fields

For a cyclic Kummer extension \(K\) of a rational function field \(k\) is considered, via class field theory, the extended Hilbert class field \(K_H^+\) of \(K\) and the corresponding extended genus field \(K_g^+\) of \(K\) over \(k\), along the lines of the definitions of R. Clement for such extens...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-08
Main Authors: Edgar Omar Curiel-Anaya, Myriam Rosalía Maldonado-Ramírez, Rzedowski-Calderón, Martha
Format: Article
Language:English
Subjects:
Field theory
Homology
Rational functions
Reciprocity
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:For a cyclic Kummer extension \(K\) of a rational function field \(k\) is considered, via class field theory, the extended Hilbert class field \(K_H^+\) of \(K\) and the corresponding extended genus field \(K_g^+\) of \(K\) over \(k\), along the lines of the definitions of R. Clement for such extensions of prime degree. We obtain \(K_g^+\) explicitly. Also, we use cohomology to determine the number of ambiguous classes and obtain a reciprocity law for \(K/k\). Finally, we present a necessary and sufficient condition for a prime of \(K\) to decompose fully in \(K_g^+\).
ISSN:2331-8422