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 $\Bbb Q.$


$\Bbb Q$ 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?


Suppose $K$ is a field extension of $F$. We say $F$ is algebraically closed in $K$ if for any polynomial $p(x) \in F[x]$ and any root of $p(x)$, $\alpha$, if $\alpha \in K$, then $\alpha \in F$. The point here is that there are no elements of $K$ algebraic over $F$ that are not already in $F$.

We say $F$ is algebraically closed if for any polynomial $p(x) \in F[x]$, $F$ contains all the roots of $p(x)$. This can be considered the most extreme form of the first definition. We are basically saying $F$ is algebraically closed in every extension of $F$, which is equivalent to saying it is algebraically closed in each extension that is a splitting field for some polynomial $p(x) \in F[x]$, i.e. every normal extension.


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