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CM fields without unit-primitive elements. (English) Zbl 1393.11072
Let \(K\) be a number field, \(O_K\) its ring of integers and \(O_K^*\) the group of units in \(O_K\). We say that \(K\) has a unit primitive element if there exists \(\theta \in O_K^*\) such that \(K=\mathbb{Q}(\theta)\). The present short note is a response to a question in [T. Zaïmi et al., Bull. Aust. Math. Soc. 93, No. 3, 420–432 (2016; Zbl 1341.11060)] concerning the following result: For every totally real field \(R\), there are infinitely many CM extensions \(K/R\) with \(R\) is the maximal totally real subfield of \(K\), without a unit primitive element. The proof is fairly straightforward. It uses a synthesis of three arguments given by R. Remak [Compos. Math. 12, 35–80 (1954; Zbl 0055.26805)]. To honour Remak’s memory, the authors tell the story of his paper and inform us that it was published posthumously more than ten years after Remak’s disappearance and presumed death in Auschwitz.

MSC:
11R27 Units and factorization
12F05 Algebraic field extensions
11R04 Algebraic numbers; rings of algebraic integers
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References:
[1] Remak, R., Über algebraische zahlkörper mit schwachem einheitsdefekt, Compositio Math., 12, 35-80, (1954) · Zbl 0055.26805
[2] Zaïmi, T.; Bertin, M. J.; Aljouiee, A. M., On number fields without a unit primitive element, Bull. Aust. Math. Soc., 93, 420-432, (2016) · Zbl 1341.11060
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