Let f(x) be an irreducible polynomial with coefficients in a field K. Then f(x) is said to be separable over K if it has no repeated zeros in a splitting field. A general polynomial f(x) in K[x] is called separable iff all its irreducible factors are.

For example, if we work over any field extension of Q (i.e. a field of characteristic zero) then all polynomials are separable. But field extensions of finite fields can have inseparable polynomials. The first of these claims follows easily from

Lemma Let f be a nonzero polynomial over some field k. Then f has a repeated zero in a splitting field of f over k iff f and df/dx have a common factor of degree > 0 in k[x].

The lemma is easily proved using the formula for differentiating a product. Bear in mind though that funny things can happen in characteristic p. For example xp-1 in Zp[x] differentiates to zero!

Finally, an algebraic field extension M of K is called separable if the minimal polynomial over K of each element of M is separable over K.

Any algebraic extension of a subfield of C is separable.