Over a perfect field, irreducible polynomials have no repeated roots.Suggest a slogan
Let \(k\) be a perfect field, for instance a field of characteristic \(0\) or a finite field. Every irreducible polynomial \(f \in k[x]\) has distinct roots in a splitting field of \(f\) over \(k\).
Over an imperfect field, an irreducible polynomial can have a repeated root: for \(k = \FF_p(t)\), the polynomial \(x^p - t\) is irreducible, and in a splitting field it factors as \((x - t^{1/p})^p\).