separable polynomial

Imagine factoring a polynomial completely over an algebraic closure, so it breaks into linear pieces. A separable polynomial is one whose roots are all distinct — none of those linear factors is repeated. Geometrically, the curve crosses the axis cleanly at each root rather than touching and bouncing back; algebraically, the polynomial does not secretly hide a double root waiting to spoil good behavior.

Precisely, a polynomial f in K[x] is separable if it has no repeated roots in an algebraic closure of K (some authors require f irreducible first). The clean test is via the derivative: f has a repeated root if and only if f and its formal derivative f' share a common factor, i.e. gcd(f, f') is nonconstant. So an irreducible f is separable exactly when f' is not zero.

Over a perfect field — every field of characteristic 0, and every finite field — every irreducible polynomial is automatically separable, so the notion only bites in positive characteristic with imperfect fields. The classic failure is x^p - t over the rational function field F_p(t): it equals (x - t^(1/p))^p, a single root repeated p times, and its derivative vanishes identically.