Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that:

$F$ is said to be separable over $\boldsymbol{k}$ if it satisfies the following equivalent conditions ($p$ denotes the characteristic exponent of $k$):

(1) $F^p$ and $k$ are linearly disjoint over $k^p$.

(2) $(k^{1/p})\otimes_k F$ is reduced.

(3) $k'\otimes_k F$ is reduced for all field estensions $k'$ of $k$.

My question is: how can I prove this equivalence? I also gave a look at separable extension, but I really don't know how to proceed maybe because I don't have understood what I need to prove.

My background: I have never attended a course in algebraic geometry and I have a basic knowledge of field theory. Moreover, any suggestion about preliminary texts I need to study is appreciated.

  • $\begingroup$ Your post does not include a question. What are you trying to do? Are you attempting to prove the equivalence of the given conditions? $\endgroup$ – KReiser Mar 25 at 1:32
  • $\begingroup$ @KReiser Noi are right. I was so tired that I forgot to include the question. Now I edit $\endgroup$ – LBJFS Mar 25 at 6:50

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