# When do algebraic closures exist constructively?

The field of algebraic numbers exists constructively, since we can represent a number by an irreducible polynomial plus an estimate in rational coordinates that separates it from any other root.

More generally, if we have a countably enumerated field with decidable arithmetic, it seems like we can construct the algebraic closure by picking a countable ordering of the irreducible polynomials, then defining an ordering of the roots of each polynomial that respects the orderings chosen for all previous polynomials.

Questions:

1. Is it correct that something like this construction works for any constructive countable field?
2. Is there a natural broader class of field for which the algebraic closure constructively exists?