| Let A1 denote the first Weyl algebra over a field K of characteristic 0; that is, A 1 is generated over K by elements p, q that satisfy the relation pq − qp = 1. One can view A1 as an algebra of differential operators by setting q = X, p = d/dX..;The basic questions which are addressed in this paper is what are all the maximal diagonalizable subalgebras of A1 and if K is not algebraically closed, what conditions should be placed on the element x ∈ A 1 so that x is diagonalizable on A 1. Thus, we use these diagonalizable elements to verify the Jacobian conjecture for n = 1. |
