I am a beginner in field theory. It seems that there is a difference between $F$ being algebraically closed in some larger extension $E$ and $F$ being algebraically closed.
For example,
$\Bbb Q$ is not algebraically closed but it is algebraically closed within $\Bbb Q.$
And
$\Bbb Q$ is algebraiclly closed within $\Bbb Q(t).$ (i.e. $\overline{\Bbb Q}_{\Bbb Q(t)}=\Bbb Q$)
I am confused with the above two examples and the difference between the two definitions. Can someone help me, please?