This is a step in Corollary 3.3.21 in Liu (Algebraic geometry and Arithmetic curve)
In the proof, we have $K/k$ a finite extension, and $k^s$ the separable closure of $k$. And we showed that $K\bigotimes_k k^s$ is a field.
The proof says
Therefore $K\bigotimes_k k^s$ is a field and it contains $(K\cap k^s)\bigotimes_k (K\cap k^s)$. It follows that $K\cap k^s=k$.
And I don't understand why.