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.