On the Structure of Elementary Nets Over Quadratic Field

Koibaev, V. A.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 4.

Abstract:
The structure of elementary nets over quadratic fields is studied. A set of additive subgroups \(\sigma=(\sigma_{ij})\), \(1\leq i,j\leq n\), of a ring \(R\) is called a net of order \(n\) over \(R\) if \(\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} \) for all \(i\), \(r\), \(j\). The same system, but without the diagonal, is called elementary ne (elementary carpet). An elementary net \(\sigma=(\sigma_{ij})\) is called irreducible if all additive subgroups \(\sigma_{ij}\) are different from zero. Let \(K=\mathbb{Q} (\sqrt{d}\,)\) be a quadratic field, \(D\) a ring of integers of the quadratic field \(K\), \(\sigma = (\sigma_{ij})\) an irreducible elementary net of order \(n\geq 3\) over \(K\), and \(\sigma_{ij}\) a \(D\)-modules. If the integer \(d\) takes one of the following values (22 fields): \(-1\), \(-2\), \(-3\), \(-7\), \(-11\), \(-19\), \(2\), \(3\), \(5\), \(6\), \(7\), \(11\), \(13\), \(17\), \(19\), \(21\), \(29\), \(33\), \(37\), \(41\), \(57\), \(73\), then for some intermediate subring \(P\), \(D\subseteq P\subseteq K\), the net \(\sigma\) is conjugated by a diagonal matrix of \(D(n, K)\) with an elementary net of ideals of the ring \(P\).

Keywords: net, carpet, elementary net, closed net, algebraic number field, quadratic field

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