Separable polynomial

In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.

This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative D P(X).

In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition In this definition, separability depended on the field K, for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use.

Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α ∈ L, which is algebraic over K, the minimal polynomial of α over K is a separable polynomial.

Inseparable extensions (that is extensions which are not separable) may occur only in characteristic p.

The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X)=0. Thus we must have

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