Definition 4

— A field K of characteristic exponent q is said to be perfect if we have $K^q = K$. When $K^q \neq K$, K is called imperfect. By this definition a field is perfect if it is of characteristic 0, or if it is a perfect ring of characteristic $p \neq 0$ in the sense of Def. 2. If $K$ is a field of characteristic $p \neq 0$ and $(K, u)$ is a perfect closure of $K$, then K is a field by Prop. 3 (V, p. 6) and $u$ is an isomorphism of K onto a subfield of $\hat{K}$. Frequently one identifies $K$ with its image under $u$ in $\hat{K}$, so that we have $\hat{K} = K^{p^{-\infty}}$ (Prop. 3). Let K be a field of characteristic 0 ; the characteristic exponent of $K$ is then 1. By convention the notation $x^{q^{-f}}$ and $S^{q^{-f}}$ is taken to mean x and S respectively (for an element $x$ of $K$ and a subset S of $K$). In particular we put $K^{q^{-\infty}} = K$ and we agree to take the perfect closure of $K$ to be $K$.