Let be a root of f. Then M is a simple extension of K such that
Every element of including is separable over K, since the irreducible polynomial of , i.e., f is separable by hypothesis.
Let be a set of separable elements over K; let be the corresponding set of irreducible polynomials of . Let be the set of all zeroes of polynomials in Y. Then K(Z) is the splitting field of polynomials in Y implying that it is separable over K. It follows that K(S), which is a subfield of K(Z), is separable over K.Deduce that if a1,...an are separable elements over K, then the extension K(a1,...,an) is separable over K.
Let be separable; let be irreducible factors of f. Then, the splitting field of Y over K is K(Z), where Z is the set of all zeroes of polynomials in Y. The splitting field of f over K, i.e., is separable over K, since f splits in its splitting field K(Z) such that , where and each has distinct roots.Conclude that if f is any separable polynomial, then the splitting field of f over K is separable over K.