# Minimal polynomial of a matrix over a field F and over an extension of F.

For example, let's make F = Q, the field of rational numbers, and K = F( $\sqrt 2$), the field obtained by adjoining $\sqrt 2$ to F (within say the field of real numbers. With n = 2, if T = (a_ij) is the matrix with a11= 1, a12= 1/2, a21= 2, a22= -1, then p= x^2 - 2, whose roots are all in K. But then p0 = p. Perhaps with n a little larger. And perhaps, after all, p0 is always equal to p.