ON TRACE FORMS OF HIGHER DEGREE

Let A be a central simple algebra over a field k, and let tr denote it s reduced trace. Then Psi(d)(X) = tr X(d) is the trace form of degree d on A. A similarity of(A Psi(d)) is a linear map f:A --> A for which there exists lambda is an element of k(x) such that Psi(d)(f(a)) = lam bda Psi(d)(a) for all a is an element of A. The ''standard'' similarit ies are the ones obtained as compositions of automorphisms, antiautomo rphisms, and scalar maps of the algebra A.