Let $[K : k]$ be a Galois extension. Show that there is a bijection between set of all $K$-linear homomorphisms $K\otimes_k K \to K$ and set of all irreducible idempotents in $K\otimes_k K$.
Here we consider $K\otimes_k K$ as $K$-algebra that acts as $a(v_1 \otimes v_2) = av_1 \otimes v_2$.
Element $e$ is called an idempotent if $e^2=e$.
There is a hint: Prove that any such homomorphism sends all but one irreducible idempotents to zero.
I can't come up with how to use it, but here is my attempt: Consider action with left multiplication by $1-e$ (if $e$ is an idempotent so does $1-e$), then it maps to zero all elements of the form $e\otimes v_2$. If $v_2 = w$ (let $w$ be another idempotent in $K$), then $e \otimes w$ is an idempotent in $K\otimes_k K$. The problems are that, firstly, I'm not sure that any arbitrarily idempotent in $K \otimes_k K$ is a tensor product of $K$ idempotents and, secondly, this map doesn't take to zero everything except chosen one. Moreover, arbitrarily homomorphism $K \otimes_k K \to K$ may not be a multiplication with some idempotent element.
Can anyone explain me how to prove this statement? Any help will be very appreciative.