Let ε be an algebraic unit for which the rank of the group of units of the order ℤ[ε] is equal to 1. Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was up to now scattered in many papers in the literature.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-14,
author = {St\'ephane R. Louboutin},
title = {Fundamental units for orders of unit rank 1 and generated by a unit},
journal = {Banach Center Publications},
volume = {108},
year = {2016},
pages = {173-189},
zbl = {06622294},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-14}
}
Stéphane R. Louboutin. Fundamental units for orders of unit rank 1 and generated by a unit. Banach Center Publications, Tome 108 (2016) pp. 173-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-14/