Defining the integers in large rings of number fields using one universal quantifier
| dc.creator | Cornelissen, Gunther | |
| dc.creator | Shlapentokh, Alexandra | |
| dc.date | 2007-08-22 | |
| dc.date | 2008-02-14 | |
| dc.date.accessioned | 2026-07-07T09:20:22Z | |
| dc.date.available | 2026-07-07T09:20:22Z | |
| dc.description | 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 $ε>0$, 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 $$\{x \in K : \ord_{\mathfrak p}x >0, \forall \mathfrak p \not \in \cal S \}$$ 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. | |
| dc.description | Substantial changes in Theorems 1 and 2 and their proofs. Two new theorems (3 and 4) | |
| dc.identifier | https://arxiv.org/abs/0708.3075 | |
| dc.identifier | http://arxiv.org/abs/0708.3075 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/154700 | |
| dc.subject | Logic | |
| dc.subject | Number Theory | |
| dc.subject | 03B25, 11U05 | |
| dc.title | Defining the integers in large rings of number fields using one universal quantifier | |
| dc.type | text |