Perfect field

Definition. A field $K$ is perfect if every algebraic extension $L/K$ is separable .

Key characterizations:

• If $\mathrm{char}(K)=0$, then $K$ is perfect.
• If $\mathrm{char}(K)=p>0$, then $K$ is perfect iff the Frobenius map $x\mapsto x^p$ is surjective (equivalently, $K^p=K$).

See also. finite fields are perfect , inseparable extension .

Examples.

1. $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are perfect (characteristic $0$).
2. Every finite field $\mathbb{F}_{p^n}$ is perfect.
3. $\mathbb{F}_p(t)$ is not perfect: $t$ is not a $p$-th power in $\mathbb{F}_p(t)$, so Frobenius is not surjective.