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 extensions of prime...

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Bibliographic Details
Main Authors Curiel-Anaya, Edgar Omar, Maldonado-Ramírez, Myriam Rosalía, Rzedowski-Calderón, Martha
Format Journal Article
LanguageEnglish
Published 30.08.2021
Subjects
Mathematics - Number Theory
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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 $K_g^+$.
DOI:10.48550/arxiv.2108.13546