Computable Fields and Arithmetically Definable Ordered Fields

A. H. Lachlan, E. W. Madison
1970 Proceedings of the American Mathematical Society
Introduction. A computable field is one whose elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are recursive. In the same vein a field is called arithmetically definable (AD for short) if its elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are arithmetical. These notions
more » ... cal. These notions clearly extend in an obvious way to ordered fields and indeed to algebraic structures in general. The term computable structure (group, ring, etc.) was probably introduced for the first time by M. 0. Rabin , however, a similar notion was discussed a few years earlier by Frohlich and Shepherdson [l]. Each of these references contains a number of interesting theorems on computable structures. Some results concerning AD structures appear in  . The main purpose of the present paper is to show that the fields of real algebraic numbers, constructible numbers, and solvable numbers, which were shown to be AD in  , are in fact computable. This answers a question raised in footnote (2) of  .