Structure of Nets over Quadratic Fields.

We study the structure of nets over quadratic fields. Let be a quadratic field, and let be the ring of integers of . A set of additive subgroups of is a net (carpet) of order over if for all values of the indices , , , . A net is irreducible if all additive subgroups are nonzero. A net is a -net if , . Let be an irreducible -net of order over , where are -modules. We prove that, up to conjugation by a diagonal matrix, all are fractional ideals of a fixed intermediate subring , , and all diagonal rings coincide with ; i.e., , where are integer ideals of for all , and if . Furthermore, for all and . [ABSTRACT FROM AUTHOR]