No finite field is algebraically closed my lecturer used this theorem in the title without proof. I didn't find a proper proof by myself. Could anyone help me out, please?

Let's suppose that $ K $ is finite and write $ K=\{\alpha_{1}, \ldots , \alpha_{n}\}$. Now take the polynomial $ p (x)=(x-\alpha_{1})\ldots (x-\alpha_{n}) +1\in K[x]$. It's easy to see that $ p (x) $ doesn't have any roots in $ K $. Hence, $ K $ is not algebraically closed.
