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# Article

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Keywords:
circular units; abelian fields; four ramified primes; Ennola relations
Summary:
In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct \$\mathbb{Z}\$-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper [2].
References:
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[2] Kučera, R., Salami, A.: Circular units of an abelian field ramified at three primes. J. Number Theory 163 (2016), 296–315. DOI:  http://dx.doi.org/https://doi.org/10.1016/j.jnt.2015.11.023 DOI 10.1016/j.jnt.2015.11.023 | MR 3459572
[3] Lettl, G.: A note on Thaine’s circular units. J. Number Theory 35 (1990), 224– 226. DOI:  http://dx.doi.org/http://dx.doi.org/10.1016/0022-314X(90)90115-8 DOI 10.1016/0022-314X(90)90115-8 | MR 1057325 | Zbl 0705.11064
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[5] Salami, A.: Bases of the group of cyclotomic units of some real abelian extension. Ph.D. thesis, Université Laval Québec, 2014.
[6] Sedláček, V.: Circular units of abelian fields. Master's thesis, Masaryk University, Faculty of Science, Brno, 2017, [online], [cit. 2017-07-17].
[7] Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math. 62 (1980/81), 181–234. DOI 10.1007/BF01389158 | MR 0595586 | Zbl 0465.12001
[8] Thaine, F.: On the ideal class groups of real abelian number fields. Ann. of Math. (2) 128 (1988), 1–18. MR 0951505 | Zbl 0665.12003

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