Orthogonal groups of nonsingular forms of higher degree

In a study [6] of the equivalence of forms, Jordan derived the finiteness of the orthogonal group of a form defined over the complex number field whose degree is at least three and whose discriminant is not zero. Herein we demonstrate the validity of Jordan’s result for any field, with the usual restriction that the characteristic of the field, if not zero, dominate the degree of the form. Jordan’s proof rests on an ingenious inequality argument applied to the coefficients of the form. Our argument begins with a proposition of Bott and Tate [3] which focuses attention on unipotent isometries. The proof provided here (due to Harrison [Sj) is regardless of the characteristic of the underlying field, as is the subsequent elimination of unipotent isometries. The argument is completed by showing the group under consideration to be periodic of bounded period with exponent relatively prime to the characteristic of the field, if it is nonzero. Finiteness then follows from a theorem of Burnside. Rather than a theorem of forms we adopt throughout the equivalent setting of symmetric multilinear maps on a vector space in the hope of achieving greater clarity of exposition and with the risk of obscuring the elementary nature of the argument. Let 0 denote a symmetric multilinear map of degree Y 2 3 on a vector space V of dimension n over a field F whose characteristic p, if nonzero, is greater than r. Let 0 = o(8) denote the orthogonal group of 0 in GL(n,F). Employing the standard extension of the scalar field to its algebraic closure, we may assume that F is algebraically closed. Recall that B is singular if and only if for some 0 # z E V, 0(X,..., z, v) = 0 for all 0 E V.