Taking $L/K$ field $I\subset L\otimes_K L$ the kernel of the multiplication $L\otimes_K L\to L$. I want to prove (if it's true?) that $L\otimes_K L$ is reduced iff $I/I^2=0$ (in the context of Kahler differential, but I'd like to prove it directly).
I can't get it: taking for example $x\otimes y\in I$ ie $xy=0$ how can I get $x\otimes y\in I^2$? $(x\otimes1)(1\otimes y)$ doesn't match because $x\otimes 1\notin I$.
Any idea?
I want to prove that because I want to prove $\Omega_{L/K}=0$ iff $L$ separable over $K$ and I know that $L$ separable over $K$ iff $L\otimes_K L$ reduced and that $\Omega_{L/K}=I/I^2$. I know there is other proof with the usual definition of Kähler differential but I'd like to use the definition $I/I^2$.