TITLE: When do algebraic closures exist constructively?
QUESTION [6 upvotes]: 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:

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

REPLY [4 votes]: This is proved in Theorem VI.3.5 of "A Course in Constructive Algebra" by Ray Mines, Fred Richman, and Wim Ruitenburg.
I'm not aware of any generalizations of this sort.