Let \(k\) be a field, \(A\in M_n(k)\), \(m_A\) the minimal polynomial of \(A\), and \(\chi_A\) its characteristic polynomial.
\(m_A\divides\chi_A\), and \(m_A\) and \(\chi_A\) have the same monic irreducible factors in \(k[x]\). Hence, if \(\chi_A=\pm\prod_{j} f_j^{a_j}\) with distinct monic irreducible \(f_j\), then \(m_A=\prod_j f_j^{b_j}\) for some \(1\leq b_j\leq a_j\), and \(m_A\) is the product of this form of least degree that annihilates \(A\).
For \(v\in k^n\), let \(p_v\) be the monic polynomial of least degree with \(p_v(A)v=0\); it is found from the first linear dependence among \(v, Av, A^2v,\ldots\). Then \(p_v\divides m_A\), and if \(e_1,\ldots,e_n\) is a basis of \(k^n\), then \(m_A=\lcm(p_{e_1},\ldots,p_{e_n})\).
By the Cayley–Hamilton theorem \(\chi_A(A)=0\), so \(m_A\divides\chi_A\). Let \(f\) be a monic irreducible factor of \(\chi_A\) and \(\lambda\in\bar k\) a root of \(f\). Then \(\lambda\) is an eigenvalue of \(A\) over \(\bar k\); if \(Av=\lambda v\) with \(v\neq0\), then \(0=m_A(A)v=m_A(\lambda)v\), so \(m_A(\lambda)=0\) and \(f\divides m_A\). Every polynomial of the stated form that annihilates \(A\) is divisible by \(m_A\), so \(m_A\) is the one of least degree.
The set \(\theset{g\in k[x]\suchthat g(A)v=0}\) is an ideal generated by \(p_v\), and it contains \(m_A\), so \(p_v\divides m_A\). Hence \(L\coloneqq\lcm_i p_{e_i}\) divides \(m_A\). Conversely \(L(A)e_i=0\) for every \(i\), so \(L(A)=0\) and \(m_A\divides L\).