Can someone verify this for me.. or tell me what reference shows me this... is this true:



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> Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \supseteq k$ the Jacobson radical of the tensor product $K\otimes\_k L$ is trivial.
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I got this idea by looking at some definitions of separable algebras (which is not my field of research.. but somehow this definition got me intruiged). Anyone knows if this is true and why so? or maybe a reference or two about it?



Characterisation for separable extension of a field
You should take a look at Theorem 3.4 (p 85) of Farb and Dennis' book *Noncommutative Algebra*. The statement is:


Let $L/k$ be a finite extension of fields. Then $K\otimes \_k L$ is semisimple for every field $K\supseteq k$ if and only if $L/k$ is a separable extension. 


That the tensor product is semisimple implies that its Jacobson radical vanishes. Conversely, any artinian ring with trivial Jacobson radical is semisimple. Therefore your equivalent formulation of separability is true.


