Separable Polynomial

In mathematics, two slightly different notions of separable polynomial are used, by different authors.

According to the most common one, a polynomial P(X) over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to its degree. In the context of polynomial factorization, such a separable polynomial is also called a square-free polynomial.

For the second definition, P(X) is separable if all of its irreducible factors in K have distinct roots in the splitting field of P(X), or equivalently in an algebraic closure of K. For this definition, separability depends explicitly on the field K, as an irreducible polynomial P which is not separable becomes separable over the splitting field of K. Also, for this definition every polynomial over a perfect field is separable, which includes in particular all fields of characteristic 0, and all finite fields.

Both definitions coincide in case P(X) is irreducible over K, which is the case used to define the notion of a separable extension of K.

In the remainder of this article, only the first definition is used.

A polynomial P(X) is separable if and only if it is coprime to its formal derivative P′(X).