On the Structure of Nets Over Quadratic Fields

Ikaev, S. S.  , Koibaev, V. A. , Likhacheva, A. O.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 3.

Abstract:
The structure of nets over quadratic fields is studied. Let \(K=\mathbb{Q} (\sqrt{d})\) be a quadratic field, \(\mathfrak{D}\) the ring of integers of the quadratic field \(K\). A set of additive subgroups \(\sigma=(\sigma_{ij})\), \(1\leq i,j\leq n\), of a~field \(K\) is called a net of order \(n\) over \(K\) if \(\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} \) for all values of the index \(i\), \(r\), \(j\). A net \(\sigma=(\sigma_{ij})\) is called irreducible if all additive subgroups \(\sigma_{ij}\) are different from zero. A net \(\sigma = (\sigma_{ij})\) is called a \(D\)-net if \(1 \in\tau_{ii}\), \(1\leq i\leq n\). Let \(\sigma = (\sigma_{ij})\) be an irreducible \(D\)-net of order \(n\geq 2\) over \(K\), where \(\sigma_{ij}\) are \(\mathfrak{D}\)-modules. We prove that, up to conjugation diagonal matrix, all \(\sigma_{ij}\) are fractional ideals of a fixed intermediate subring \(P\), \(\mathfrak{D}\subseteq P \subseteq K\), and all diagonal rings coincide with \(P\): \(\sigma_{11}=\sigma_{22}=\ldots =\sigma_{nn}=P,\) where \(\sigma_{ij}\subseteq P\) are integer ideals of the ring \(P\) for any \(i < j\), if \(i > j\), then \(P\subseteq\sigma_{ij}\). For any \(i\), \(j\) we have \(\sigma_{1j}\subseteq\sigma_{ij}\).

Keywords: nets, carpets, algebraic number field, quadratic field

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