## 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
Full Text:

### References:

 [1] R. Deheuvels : Formes quadratiques et groupes classiques , P.U.F. - Paris - 1981 . MR 657579 | Zbl 0467.15015 · Zbl 0467.15015 [2] J. Dieudonne : Sur les groupes classiques Hermann - Paris - 1973 . MR 344355 | Zbl 0926.20030 · Zbl 0926.20030 [3] S.H. Friedberg : Extending the principal axis theorem to fields other than R, Ann. Math. Monthly - vol 97 , n^\circ 2 - 1990 - p. 147 - 149 . MR 1041894 | Zbl 0735.15007 · Zbl 0735.15007 [4] P. Ribenboim : L’arithmétique des corps , Hermann - collection Méthodes - Paris - 1972 . MR 330093 | Zbl 0253.12101 · Zbl 0253.12101 [5] J.P. Serre : Cours d’arithmétique , P.U.F. - Le Mathématicien - 1970 . Zbl 0225.12002 · Zbl 0225.12002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.