I have a finite extension of domains $A\hookrightarrow B$ which induce a finite field extension $k:=\operatorname{Frac}(A)\hookrightarrow\operatorname{Frac}(B)=:K$. How can I prove $K=B\otimes_Ak$ (isomorphism as vector spaces over $k$).
Clearly I have that the right-hand-module is contained in the left, but why is there an equality?
Thank you!!