2007/08/22 by Gunther Cornelissen, Cornelissen, Gunther, Alexandra Shlapentokh +1
Computer Science · Mathematics · #03B25 #11U05 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #math.LO #math.NT #msc:03B25 #msc:11U05
paper · pdf · doi:10.48550/arxiv.0708.3075
Substantial changes in Theorems 1 and 2 and their proofs. Two new theorems (3 and 4)
openalex publication_date 2007/08/22 · arxiv created 2008/02/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Julia Robinson has given a first-order definition of the rational integers \mathbb Z in the rational numbers \mathbb Q by a formula (∀ ∃ ∀ ∃)(F=0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including \mathbb Q, for every ε>0, there exists a set of primes \cal S of natural density exceeding 1-ε, such that \mathbb Z can be defined as a subset of the ``large'' subring \x ∈ K : \ord\mathfrak px >0, ∀ \mathfrak p \not ∈ \cal S \ of K by a formula of the form (∃ ∀ ∃)(F=0) where there is only one ∀-quantifier, and where F is a polynomial.