Separable polynomial

A polynomial over a field is separable when it has no repeated root in a splitting field. Equivalently, it is coprime to its formal derivative.

A nonzero polynomial is factorwise separable if every irreducible factor is a separable polynomial. Unlike the convention that all roots must be distinct, this condition allows repeated irreducible factors. Its splitting field is a separable field extension precisely when the polynomial is factorwise separable: in one direction every root has a separable minimal polynomial; in the other, adjoining finitely many roots with separable minimal polynomials gives a separable extension. Thus the implication from a separable splitting extension to a polynomial with distinct roots needs the extra squarefree hypothesis. The example $(X-1{)}^2$ already distinguishes the conventions.

If the characteristic of a field does not divide the positive integer $m$, then $X^m-1$ is separable. Its formal derivative is $m{X}^{m-1}$, and a common root would have to be both nonzero and zero.

If $f$ is irreducible over a field of characteristic zero, then $f^{\prime }\ne 0$ and $\mathrm{deg}{f}^{\prime }<\mathrm{deg}f$. Thus $f$ cannot divide $f^{\prime }$, so irreducibility gives $\gcd (f,f^{\prime })=1$. Consequently $f$ has no repeated root in any extension field.