I had forgotten that Vandermonde matrices diagonalize Companion matrices. (Probably because of all the barnacles that came with undergrad linear algebra.) Oh, and determinant(Vandermonde) = sqrt(discriminant of polynomial) (sort of ... modulo signs) Yet another way to see that solving the polynomial requires a sqrt. (%i1) C:polytocompanion((x-r1)*(x-r2)*(x-r3),x),factor; [ 0 0 r1 r2 r3 ] [ ] (%o1) [ 1 0 - (r2 r3 + r1 r3 + r1 r2) ] [ ] [ 0 1 r3 + r2 + r1 ] (%i2) V:vandermonde_matrix([r1,r2,r3]); [ 2 ] [ 1 r1 r1 ] [ ] (%o2) [ 2 ] [ 1 r2 r2 ] [ ] [ 2 ] [ 1 r3 r3 ] (%i3) D:diag_matrix(r1,r2,r3); [ r1 0 0 ] [ ] (%o3) [ 0 r2 0 ] [ ] [ 0 0 r3 ] (%i4) (V^^-1).D.V,factor; [ 0 0 r1 r2 r3 ] [ ] (%o4) [ 1 0 - (r2 r3 + r1 r3 + r1 r2) ] [ ] [ 0 1 r3 + r2 + r1 ] (%i5) %-C,expand; [ 0 0 0 ] [ ] (%o5) [ 0 0 0 ] [ ] [ 0 0 0 ] (%i6)