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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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
30.08.2021
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Subjects | |
Online Access | Get full text |
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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 $K_g^+$. |
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DOI: | 10.48550/arxiv.2108.13546 |