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On Ochoa's special matrices in matrix classes
Authors:Hans P Rehm
Institution:Math. Institut II der Universität 7500 Karlsruhe 1 Englerstr. 2 Federal Republic of Germany
Abstract:It is well known that the ideal classes of an order Zμ], generated over Z by the integral algebraic number μ, are in a bijective correspondence with certain matrix classes, that is, classes of unimodularly equivalent matrices with rational integer coefficients. If the degree of μ is ?3, we construct explicitly a particularly simple ideal matrix for an ideal which is a product of different prime ideals of degree 1. We obtain the following special n×n matrix (cij) in the matrix class corresponding to the ideal class of our ideal: ci+1,i=1(i=1,…,n?2); cij=0(?i?n, 1?j?n? 2, and ij+1); cnj=0(j)=2,…,n?1). The remaining coefficients are given as explicit polynomials in an integer z which depends on the ideal. It is shown that the matrix class of every regular ideal class of Zμ] contains a special matrix of this kind.
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