Proposition 10.157.9. Let $k \subset K$ be a field extension. If the characteristic of $k$ is zero then

$K$ is separable over $k$,

$K$ is geometrically reduced over $k$,

$K$ is formally smooth over $k$,

$H_1(L_{K/k}) = 0$, and

the map $K \otimes _ k \Omega _{k/\mathbf{Z}} \to \Omega _{K/\mathbf{Z}}$ is injective.

If the characteristic of $k$ is $p > 0$, then the following are equivalent:

$K$ is separable over $k$,

the ring $K \otimes _ k k^{1/p}$ is reduced,

$K$ is geometrically reduced over $k$,

the map $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K/\mathbf{F}_ p}$ is injective,

$H_1(L_{K/k}) = 0$, and

$K$ is formally smooth over $k$.

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