The multiplicative structure of the ring of integers \(O_K\) of a number field \(K\) carries important information about the field, of course. However, the additive semigroup \(O_K^+\) of ‘totally positive’ elements of \(O_K\) also plays a significant role when \(K\) is a totally real number field. The paper under review studies the case of real quadratic fields \(K = \mathbb{Q}(\sqrt{D})\). Here, it is natural to characterize various properties of \(O_K^+\) in terms of the continued fraction expansion of \(\sqrt{D}\).
The authors first obtain a description of the elements of \(O_K^+\) for \(K = \mathbb{Q}(\sqrt{D})\) which are indecomposable into two or more elements of \(O_K^+\), in terms of the continued fraction expansion of \(\sqrt{D}\). The indecomposable elements of \(O_K^+\) form a generating set for it and they have relations among them. In order to determine the relations between indecomposable elements, the authors first prove the interesting key result that every totally positive element of \(O_K\) is a positive linear combination of two consecutive indecomposable totally positive elements. Using this, they deduce all the relations neatly in terms of the continued fraction expansion of \(\sqrt{D}\). As uniqueness of decomposition of elements of \(O_K^+\) as a sum of indecomposables cannot be expected to hold (as there are relations among the indecomposables), the best one can hope for is to characterize those totally positive elements which are uniquely decomposable into indecomposables. The authors accomplish this again via the continued fraction expansion of \(\sqrt{D}\). Finally, using this last-mentioned result, the authors are able to deduce the remarkable theorem that the additive semigroup \(O_K^+\) determines the continued fraction of \(\sqrt{D}\) and hence, the field \(K\) up to isomorphism. This is striking because the ring of integers of any quadratic field does not determine the field at all from its additive group structure, as \(O_K \cong\mathbb{Z}^2\) as an abelian group.