which has as a root. among all polynomials in which have as root choose one with smallest degree. call it we may assume that is monic, i.e. the leading coefficient of is
1, because, if not, then we could replace by where is the leading coefficient of now if is another polynomial which has as a root, then by Euclidean algorithm there exist
in such that either or and but then so we can't have that because we
chose to be a polynomial with smallest degree satisfying the condition so there's only one option left for which is thus i.e.
finally is unique, i.e. if is monic, and then this is because, by what we just proved, we must have for some
but since we must have i.e. a constant. so but then because both are monic. now we can give a
name: we call it " the minimal polynomial of over ". note that the same result is true if we replace by any field.