Remark

— 1) When $E$ is algebraic and of infinite degree over $K$, the condition $E = K(E^p)$ does not always ensure that $E$ is separable over $K$. For example, if $K$ is imperfect and $E$ is a perfect closure of $K$, then we have $E = K(E^p)$ but $E$ is not a separable extension of $K$ ($V$, p. 39, Cor. 3). 2) Let $E$ a separable algebraic extension of a field $K$ of characteristic exponent $p$. Then we have $E^p \cap K = K^p$ (Cor. 2); therefore if $E$ is perfect, the same is true of $K$.