is a field. If contains an inseparable polynomial, then can not have characteristic 0 or is some finite field with characteristic .

In general, we do not need to worry about the separability of the extension field, because we often deal with finite fields or the one of characteristic 0.

**An example of inseparable polynomial**

Let be a field of characteristic , and is the field of rational polynomials, is the indeterminate. Consider , y is the indeterminate, is an irreducible element in this UFD. So is irreducible in by the Einsenstein criterion. There exists a splitting field of over . Suppose is root of then . By freshman’s dream . So has multiple roots. It is not separable.

Isaacs, p281