Number Fields Whose Integers Are Sums of Units

Definitions and mathematical statement

For a number field $K$, that is, a finite extension of ${\mathbb{Q}}$, let $\mathcal O_K$ be its ring of integers and let $\mathcal O_K^\times$ be its group of units. Determine all such fields $K$ for which \[ \forall\alpha\in\mathcal O_K\quad\exists m\in{\mathbb{Z}}_{\geq0}\quad\exists u_1,\ldots,u_m\in\mathcal O_K^\times \quad \alpha=\sum_{i=1}^m u_i. \] The number of summands may depend on $\alpha$, and units may be repeated. The empty sum represents zero; allowing only positive lengths gives the same property because $0=1+(-1)$. Equivalently, the additive group of $\mathcal O_K$ is generated by its units, since the negative of a unit is a unit. The request is a classification of number fields, naturally up to field isomorphism, with this property. It does not require a uniform bound on the number of summands.

Short English statement

Classify the number fields whose every algebraic integer is a finite sum of units of their ring of integers. The length of the sum may vary with the integer.

Sources

• Local UnsolvedMath registry, permanent ID $30000751$, source code catalogue entry, "Number Fields Whose Integers Are Sums of Units." No individual proposer is supplied.
• Diophantische Approximationen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/21, pp. 1115--1190: https://doi.org/10.4171/owr/2007/21.
• Supplied literature source, "survey of additive unit representations": https://pmc.ncbi.nlm.nih.gov/articles/PMC4370826/.
• The supplied literature triage, checked 2026-08-21, labels the classification open. This dated assessment and the contents of its references have not been independently verified; the draft does not certify current status.