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Remarques sur les matrices orthogonales (resp. symétriques) à coefficients p-adiques. (Remarks on orthogonal (respectively symmetric) matrices with p-adic coefficients). (French) Zbl 0731.15019

Let K be a field of characteristic different from 2, and let E be an n- dimensional vector space over K on which is defined a symmetric bilinear form f and the corresponding quadratic form q. If V is a subspace let \(V^{\perp}\) denote the set of all x such that \(f(x,y)=0\) for all \(y\in V\). If \(V\subset V^{\perp}\) then we say V is totally isotropic. The maximal totally isotropic subspaces all have the same dimension which is called the index of q.
The author studies the case where K is a field of p-adic numbers, \(E=K^ n\), and q is the standard quadratic form \(q(x)=x^ 2_ 1+...+x^ 2_ n\). In this case he obtains a value for the index as a function of n and the residue class of p mod 4.
The author then turns to the study of orthogonal and symmetric matrices over such fields. He obtains results concerning when such matrices can or cannot be diagonalized.

MSC:

15A63 Quadratic and bilinear forms, inner products
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices
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References:

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