The Stacks project

Proposition 10.158.9. Let $K/k$ be a field extension. If the characteristic of $k$ is zero then

  1. $K$ is separable over $k$,

  2. $K$ is geometrically reduced over $k$,

  3. $K$ is formally smooth over $k$,

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

  5. 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:

  1. $K$ is separable over $k$,

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

  3. $K$ is geometrically reduced over $k$,

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

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

  6. $K$ is formally smooth over $k$.

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