# Equivalent definitions of separable extension of a field

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.

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