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On the compositum of orthogonal cyclic fields of the same odd prime degree

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  14 July 2020

Cornelius Greither  and
Radan Kučera
Cornelius Greither
Affiliation:
Universität der Bundeswehr München, Neubiberg, Germany e-mail: cornelius.greither@unibw.de
Radan Kučera*
Affiliation:
Masaryk University, Brno, Czech Republic
*
e-mail: kucera@math.muni.cz
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Abstract

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

Keywords

Circular (cyclotomic) unitsabsolutely abelian fieldsclass groups

MSC classification

Primary: 11R20: Other abelian and metabelian extensions
Secondary: 11R27: Units and factorization
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 6 , December 2021 , pp. 1506 - 1530
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

R. K. was supported by Project 18-11473S of the Czech Science Foundation.

References

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