A compositum is separable exactly when both constituent algebraic extensions are separable.Suggest a slogan
Let \(k\) be a field and \(E, F\) algebraic extensions of \(k\) contained in a common field, with compositum \(EF\). Then \(E/k\) and \(F/k\) are separable if and only if \(EF/k\) is separable.