Defining the integers in large rings of number fields using one universal quantifier

Details Defining the integers in large rings of number fields using one universal quantifier Defining the integers in large rings of number fields using one universal quantifier

2007-08-22

arxiv.org/abs/0708.3075v2

Mathematics

Logic

Substantial changes in Theorems 1 and 2 and their proofs. Two new theorems (3 and 4)

Scientific paper

Julia Robinson has given a first-order definition of the rational integers $\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-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 $\epsilon>0$, there exists a set of primes $\cal S$ of natural density exceeding $1-\epsilon$, such that $\mathbb Z$ can be defined as a subset of the ``large'' subring $$\{x \in K : \ord_{\mathfrak p}x >0, \forall \mathfrak p \not \in \cal S \}$$ of K by a formula of the form $(\exists \forall \exists)(F=0)$ where there is only one $\forall$-quantifier, and where F is a polynomial.

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