Classification theorems for central simple algebras with involution

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces a re adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian f orms. Analogues of the classical invariants of quadratic forms (discriminan t, Clifford algebra, signature) have been defined for arbitrary central sim ple algebras with involution. In this paper it is shown that over certain f ields these invariants are sufficient to classify involutions up to conjuga tion. For algebras of low degree a classification is obtained over an arbit rary field.