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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?
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  1. This is proved in Theorem VI.3.5 of "A Course in Constructive Algebra" by Ray Mines, Fred Richman, and Wim Ruitenburg.
  2. I'm not aware of any generalizations of this sort.
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  • $\begingroup$ Would it generalize to any well ordered field? $\endgroup$ Jun 21 '20 at 21:48
  • $\begingroup$ It seems so, but I'm not sure. $\endgroup$ Jun 21 '20 at 21:56

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