[math-fun] Expansion of charpoly's in terms of traces ?

I noticed in Wikipedia's description of eigenvalues, an expansion of the characteristic polynomial of a 3x3 matrix in terms of traces, so I wanted to see if this worked more generally. For a 2x2 matrix, the charpoly is: x^2-x*tr(M)+det(A) = x^2-x*tr(M)+ (tr(M)^2-tr(M^2))/2 For a 3x3 matrix, the charpoly is: x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-det(M) = x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6 For a 4x4 matrix, the charpoly is: x^4-x^3*tr(M)+x^2*(tr(M)^2-tr(M^2))/2-x*(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6+det(M) Notice that the higher order terms are the same form even though the dimension is higher. Can we continue this indefinitely? What is the general term?