Being integral over a base ring generalizes being algebraic over a base field. However, it seems to me that the terminology surronding these two notions doesn't quite align. See here, for example, for a definition of integrally closed subring.
In particular, it seems to me that if we consider $\mathbb{R}$ as a field extension of itself, then this is an integrally closed extension (because every element $a$ of $\mathbb{R}$ can be killed by a monic polynomial with coefficients in $\mathbb{R}$, namely $x-a$), but it's not an algebraically closed extension (because the larger field fails to be algebraically closed.)
Question. Am I right to say that "integrally closed extension" and "algebraically closed extension" don't agree in their meaning, even for field extensions? If so, what's the reason for this terminological discrepancy? Is it just historical inertia, or is there a deeper reason for the difference?
Best Answer
If $L/k$ is a field extension, we say that $k$ is algebraically closed in $L$ if every element in $L$ that is algebraic over $k$ is already contained in $k$. This is precisely the same as $k$ being integrally closed in $L$. But this is not to be confused with the notion of an algebraically closed field, being a field that does not admit any nontrivial algebraic extensions.
So you are right that sometimes we call an extension algebraically closed if the larger field is algebraically closed, not if the base field is algebraically closed inside the larger field. This is perhaps an unfortunate use of words.