The algebraic closure of the field of real numbers is the field of complex numbers.
The algebraic closure of the field of rational numbers is the field of algebraic numbers.
The completion of the set of rational numbers is the set of the real numbers, in the sense of either order (Dedekind completion) or metric.
I was wondering if the completion of the set of algebraic numbers in the sense of either order (what kind of order is it?) or metric is the set of complex numbers?
Thanks and regards!