On Number Fields Without a Unit Primitive Element

Abstract

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

Cm Fields Without Unit-Primitive Elements

“…We thank Marie José Bertin for her kind and helpful correspondence, and we hope that these short remarks fulfil their double purpose: explaining the fairly straightforward solution of Question 1.8 in [2] and honouring Remak's memory.…”

“…This note is an addendum to the recent paper [2] by Zaïmi, Bertin and Aljouiee. Recall that a number field K is said to admit a unit primitive element (UPE) if there exists a unit θ ∈ O * K such that K = Q(θ).…”

“…Now for the story. The argument given above is a synthesis of three (closely related) arguments: one by Zaïmi, one by C. Greither (found while writing a review of [2] for Mathematical Reviews), and, last but not least, one by Robert Remak in [1]. The existence of the latter argument was discovered and communicated to Marie José Bertin by W. Narkiewicz after Zaïmi had found his proof.…”

“…(Here, as usual, O * K is the group of units in the ring O K of integers of K.) Recall also that every non-CM field admits a UPE. The following question (labelled 1.8) was asked in [2]: Given any totally real field R, does there exist a CM field K (or even infinitely many such fields) which has no UPE and for which R is equal to K + , the maximal totally real subfield of K?…”

CM fields with a reciprocal unit-primitive element

“…Several authors studied the question, initiated by Lalande, in [5], of whether a number field admits a reciprocal unitprimitive element (see, for instance, [4], [7], [8] and the references therein). An algebraic number ϑ is said to be reciprocal if 1/ϑ is a conjugate of ϑ .…”