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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

Greither, C. and Kučera, R., Annihilators for the class group of a cyclic field of prime power degree . Acta Arith. 112(2004), 177–198. http://dx.doi.org/10.4064/aa112-2-6 CrossRefGoogle Scholar
Greither, C. and Kučera, R., Annihilators for the class group of a cyclic field of prime power degree III . Publ. Math. 86(2015), 401–421. http://dx.doi.org/10.5486/PMD.2015.7029 Google Scholar
Greither, C. and Kučera, R., Linear forms on Sinnott’s module . J. Number Theory 141(2014), 324–342. http://dx.doi.org/10.1016/j.jnt.2014.02.003 CrossRefGoogle Scholar
Greither, C. and Kučera, R., Washington units, semispecial units, and annihilation of class groups. Manuscripta Math. 2020. http://dx.doi.org/10.1007/s00229-020-01241-y CrossRefGoogle Scholar
Herman, J., Annihilators of the class group of a compositum of quadratic fields . Arch. Math. (Brno) 49(2013), 209–222. http://dx.doi.org/10.5817/AM2013-4-209 CrossRefGoogle Scholar
Kučera, R., On the class number of a compositum of real quadratic fields: an approach via circular units . Funct. Approx. Comment. Math. 39(2008), 179–189. http://dx.doi.org/10.7169/facm/1229696569 CrossRefGoogle Scholar
Lettl, G., A note on Thaine’s circular units . J. Number Theory 35(1990), 224–226. http://dx.doi.org/10.1016/0022-314X(90)90115-8 CrossRefGoogle Scholar
Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field . Invent. Math. 62(1980), 181–234. http://dx.doi.org/10.1007/BF01389158 CrossRefGoogle Scholar