Let K be a field and f a member of K[x] a separable polynomial. Prove that the simple extension M= K[x]/(f) (where (f) is the ideal generated by f) is separable over K. Deduce that if a1,...an are separable elements over K, then the extension K(a1,...,an) is separable over K. Conclude that if f is any separable polynomial, then the splitting field of f over K is separable over K.

Any ideas? I think for the first part I should be using a theorem about the number of homomorphisms from M to a splitting field, but I don't quite see how it fits in...

Many thanks.