I am taking 4th field theory, and I am trying to solve a problem:
Find a counter example for the following false statement: If $K$ is algebraically closed and $F \subseteq K$ is a subfield of $K$, then $K$ is an algebraic closure of $F$.
I tried many examples, like $\mathbb{Q}(i)/\mathbb{Q}$, $\mathbb{C}/\mathbb{R}$, but none of them satisfy the requirement. Can somebody help me cook up a counter example? Thanks!