Perfect field

Let $F$ be a field .

Definition (perfect field). The field $F$ is perfect if every algebraic field extension of $F$ is a separable extension .

There are standard equivalent characterizations:

• $F$ is perfect iff every algebraic element over $F$ is a separable element .
• If $\mathrm{char}(F)=0$, then $F$ is perfect.
• If $\mathrm{char}(F)=p>0$, then $F$ is perfect iff the Frobenius endomorphism $\varphi:F\to F$, $\varphi(a)=a^p$, is surjective (equivalently $F=F^p$).

Perfect fields are precisely the base fields over which “separable vs. algebraic” coincide: every algebraic extension automatically has no inseparable phenomena.

Examples.

1. $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are perfect because they have characteristic $0$.
2. Every finite field $\mathbb{F}_{p^n}$ is perfect (in characteristic $p$, Frobenius is automatically bijective on a finite set).
3. $\mathbb{F}_p(t)$ is not perfect: Frobenius is not surjective since $t\notin (\mathbb{F}_p(t))^p$. Equivalently, the extension $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$ is inseparable .