Weakly integrally closed domains:minimum polynomials of matrices

Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is ’yes’ for 3-by-3 matrices. It is shown that certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers.