Ž . JOURNAL OF ALGEBRA 196, 283300 1997 ARTICLE NO. JA977071 Symmetric Composition Algebras Alberto Elduque* Departamento de Matematicas y Computacion, Uni  ersidad de La Rioja, ´ ´ 26004, Logrono, Spain ˜ Communicated by Efim Zelmano  Received December 18, 1996 A construction of all the Okubo algebras over fields of characteristic 3 is provided and the complete classification of the composition algebras with associa- tive norm over fields is finished.  1997 Academic Press 1. INTRODUCTION An algebra A over a field F is said to be a composition algebra if there Ž . exists a strictly nondegenerate quadratic form called the norm n: A  F such that n xy  n xn y 1 Ž . Ž . Ž . Ž. for any x, y  A. The form being strictly nondegenerate means that its polarization, defined by n x , y  n x  y  n x  n y , Ž . Ž . Ž . Ž . is a nondegenerate bilinear form. Ž. Linearization of 1 gives n xy , xz  n yx , zx  n xn y , z , Ž . Ž . Ž . Ž . 2 Ž. n xy , zt  n zy , xt  n x , zn y , t , Ž . Ž . Ž . Ž . for any x, y, z, t  A. Ž . *Supported by the Spanish DGICYT PB 94-1311-C03-03 and by the Universidad de Ž . La Rioja Convenio Caja Rioja, 95PYB06MBC and 96PYB07PBC . Present address: Departa- mento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain. E-mail: ´ elduque@posta.unizar.es. 283 0021-869397 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.