Let \(L/K\) be an algebraic field extension. \(L/K\) is separable if for every \(\alpha \in L\) the minimal polynomial \(m_\alpha\in K[x]\) of \(\alpha\) over \(K\) is separable.
Let \(K\) be a field and \(f\in K[x]\) irreducible. Then \(f\) is separable if and only if its formal derivative \(f'\) is nonzero.
Let \(\bar K\) be an algebraic closure of \(K\). A root \(\beta\in\bar K\) of \(f\) is repeated if and only if \(f'(\beta)=0\), since writing \(f=(x-\beta)g\) gives \(f'(\beta)=g(\beta)\). If \(f'=0\), every root of \(f\) is repeated, so \(f\) is not separable. If \(f'\neq0\) and \(\beta\) is a repeated root, then \(f\), being irreducible, is a scalar multiple of the minimal polynomial of \(\beta\) over \(K\), so \(f\) divides \(f'\); this is impossible because \(\deg f'<\deg f\) and \(f'\neq0\).