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# perfect field

A *perfect field* is a field $K$ such that every algebraic extension field $L/K$ is separable over $K$.

All fields of characteristic 0 are perfect, so in particular the fields $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Q}$ are perfect. If $K$ is a field of characteristic $p$ (with $p$ a prime number), then $K$ is perfect if and only if the Frobenius endomorphism $F$ on $K$, defined by

$F(x)=x^{p}\quad(x\in K),$ |

is an automorphism of $K$. Since the Frobenius map is always injective, it is sufficient to check whether $F$ is surjective. In particular, all finite fields are perfect (any injective endomorphism is also surjective). Moreover, any field whose characteristic is nonzero that is algebraic over its prime subfield is perfect. Thus, the only fields that are not perfect are those whose characteristic is nonzero and are transcendental over their prime subfield.

Similarly, a ring $R$ of characteristic $p$ is perfect if the endomorphism $x\mapsto x^{p}$ of $R$ is an automorphism (i.e., is surjective).

## Mathematics Subject Classification

12F10*no label found*

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