Model-theoretic algebraic closure in non-algebraically closed field

Question

It is well-known that if $M$ is an algebraically closed field, then the model-theoretic algebraic closure of any set $A \subset dom(M)$ is the same as its algebraic closure in the sense of field theory (roots of polynomials in $A[x]$). Does the same hold if the field is not algebraically closed? I need a proof or a counterexample.

Answer 1

There are certainly fields which are not algebraically closed, but in which model-theoretic algebraic closure agrees with relative field-theoretic algebraic closure. For example, this happens in real closed fields, p-adically closed fields, and pseudofinite fields.

On the other hand, this is not true in all fields. Many examples are given / constructed in the following paper: https://www.jstor.org/stable/3648549