Revisions to Characterisation for separable extension of a field

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edited Jan 20, 2014 at 13:35

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

Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \supset k$$L \supseteq k$ the Jacobson radical of the tensor product $K\otimes_k L$ is trivial.

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?

edited tags and texified the question

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edit approved Jan 20, 2014 at 13:35

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

Let k be a field then a field extension K of k is separable over k iff

for any field extension L >= k the Jacobson radical of the tensor product K (x)_k L is trivial.

Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \supset k$ the Jacobson radical of the tensor product $K\otimes_k L$ is trivial.

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?