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**A note on Thaine’s circular units.**
*(English)*
Zbl 0705.11064

F. Thaine [Ann. Math., II. Ser. 128, No.1, 1-18 (1988; Zbl 0665.12003)] defined the circular units of an abelian field K (with conductor m) by means of the polynomials
\[
f_ j(X)=\pm \prod^{j}_{i=1}\prod^{m-1}_{k=1}(X^ i-\zeta^ k_ m)^{a_{i,k}}\in K[X],
\]
where \(\zeta_ m\) is a primitive mth root of 1 and \(a_{i,k}\) are rational integers. Thaine’s group G of circular units in fact consists of all those numbers \(f_ j(1)\), for \(j\geq 1\), which are units. The present author proves that G equals the group of circular units introduced by W. Sinnott [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. He also shows that one obtains the whole group G by taking the polynomials \(f_ j(X)\) for \(j=1\) only.

Reviewer: T.Metsänkylä

### References:

[1] | Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., 62, 181-234 (1980) · Zbl 0465.12001 |

[2] | Thaine, F., On the ideal class groups of real abelian number fields, Ann. of Math., 128, 1-18 (1988) · Zbl 0665.12003 |

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