By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, in the language of 0,1, + and $\times$, is recursive. Is this field either known to be decidable, or known not to be?


As of 1963 Tarski's question of whether this field is decidable was open -- so i doubt any simple adaptation of his result on real closed fields can settle this question. He conjectured that the only decidable fields were finite, real closed, or algebraically closed. See Julia Robinson, The decision problem for fields, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 299–311. especially pages 302 and 305. 


Much has gone on since 1963, and Tarski's general conjecture is well refuted, but I do not find a solution to this problem.



Is the field of constructible numbers known to be decidable?
According to the following paper:


Carlos R. Videla, On the constructible numbers, Proceedings of the American Mathematical Society Vol. 127, No. 3 (Mar., 1999), pp. 851-860.


The problem has remained open at least until 1999. I think the problem is still open. In the above paper the author proves that the ring of constructible algebraic integers is first-order definable in the field of constructible numbers. The author hopes that $\mathbb{Z}$ should be definable in the ring of constructible algebraic integers and therefore his result would be a partial result towards resolving the problem negatively.


