Rumely's Local Global Principle for Algebraic PSC Fields over Rings

• Published in: Transactions of the American Mathematical Society 1998-01, Vol.350 (1), p.55-85
• Main Authors: Jarden, Moshe, Razon, Aharon
• Format: Article
• Language: English
• Subjects: Algebra; Approximation; Cantors theorem; Coordinate systems; Existence theorems; Integers; Mathematical functions; Mathematical rings; Mathematical theorems
• ISSN: 0002-9947
• DOI: 10.1090/s0002-9947-98-01630-4

Let S be a finite set of rational primes. We denote the maximal Galois extension of Q in which all p∈ S totally decompose by N. We also denote the fixed field in N of e elements σ1,...,σein the absolute Galois group G(Q) of Q by N(σ). We denote the ring of integers of a given algebraic extension M of Q by ZM. We also denote the set of all valuations of M (resp., which lie over S) by VM(resp., SM). If v∈ VM, then OM,vdenotes the ring of integers of a Hanselization of M with respect to v. We prove that for almost all σ ∈ G(Q)e, the field M=N(σ) satisfies the following local global principle: Let V be an affine absolutely irreducible variety defined over M. Suppose that$V(O_{M,v})\neq \varnothing $for each$v\in \scr{V}_{M}\backslash S_{M}$and$V_{\text{sim}}(O_{M,v})\neq \varnothing $for each v∈ SM. Then$V(O_{M})\neq \varnothing $. We also prove two approximation theorems for M.