Converting Definability to Undecidability over infinite extensions
For certain totally real fields one can easily convert definability results into undecidability results. A result of J. Robinson implies that if a ring of integers has a certain invariant which C. Videla called a “Julia Robinson number”, one can define a first-order model of Z over the ring. The Julia Robinson number s of a ring R of totally real integers is a real number s or to, such that (0, s) is the smallest interval containing infinitely many sets of conjugates of numbers of R, i.e., infinitely many x e R with all the conjugates (over Q) in (0, s). A result of Kronecker implies that s > 4 (see ), and therefore if a totally real ring of integers in question contains the real parts of infinitely many distinct roots of unity, the Julia Robinson number for the ring is indeed 4, and we have the desired undecidability result. Thus using our definability results we can conclude that for any fixed rational prime q and positive integer m the elementary theory of the largest totally real subfield of the cyclotomic field Q(^tn, n e Z>0, t — 1 ф mod qm) is undecidable.
One can also obtain undecidability results for elementary theory of fields where we know the integral closure of some rings of S-integers to be existentially undecidable. (See [51, 57]). With the help of these old existential undecidability result we obtain the following theorem.
Theorem 3.12 Rational numbers are first-order definable in any abelian extension of Q with finitely many ramified primes, and therefore the first-order theory of such afield is undecidable.