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Genus fields of Kummer $ℓ^{n}$ -cyclic extensions

Carlos Daniel Reyes-Morales

Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del I.P.N., Mexico

E-mail Address: mcenigm@gmail.com

E-mail Address: creyes@ctrl.cinvestav.mx

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and

Gabriel Villa-Salvador

Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del I.P.N., Mexico

E-mail Address: gvillasalvador@gmail.com

E-mail Address: gvilla@ctrl.cinvestav.mx

Corresponding author.

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https://doi.org/10.1142/S0129167X21500622Cited by:2 (Source: Crossref)

Abstract

We give a construction of the genus field for Kummer $ℓ^{n}$ -cyclic extensions of rational congruence function fields, where $ℓ$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $ℓ$ -cyclic extension. Finally, we study the extension ${(K_{1} K_{2})}_{𝔤 𝔢} / {(K_{1})}_{𝔤 𝔢} {(K_{2})}_{𝔤 𝔢}$ , for $K_{1}$ , $K_{2}$ abelian extensions of $k$ .

Communicated by Akihiko Yukie

Keywords:

AMSC: 11R60, 11R29, 11R58

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