Definability of the ring of integers in some infinite algebraic extensions of the rationals

摘要

Let K be an infinite Galois extension of the rationals such that every finite subextension has odd degree over the rationals and its prime ideals dividing 2 are unramified. We show that its ring of integers is first-order definable in K. As an application we prove that together with all its Galois subextensions are undecidable, where Δ is the set of all the prime integers which are congruent to -1 modulo 4.