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 *x*^{p}-1
in **Z**_{p}*[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.