minimal polynomial

Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial divides the characteristic polynomial. Also, any root of the characteristic polynomial is also a root of the minimal polynomial, so the two kinds of polynomial have the same roots, only (possibly) differing in their multiplicities.)

[…] we are given an element 𝛼∈E, and want to compute the minimal polynomial 𝜙∈F[X] of 𝛼 over F.