Rumely's local global principle for algebraic psc fields over rings

Moshe Jarden, Aharon Razon

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a finite set of rational primes. We denote the maximal Galois extension of Q in which all p ε S totally decompose by N. We also denote the fixed field in N of e elements σ1,... ,εe in 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 ε VM then OM, denotes the ring of integers of a Henselization of A/ with respect to v. We prove that for almost all σ ε G(Q)e, the field M = N(cr) satisfies the following local global principle: Let V be an affine absolutely irreducible variety defined over M. Suppose that V(OM,v) ≠ Ø for each v ε VM\SM and Vsim(OM,v) ≠ Ø for each v ε SM. Then V(OM) ≠ Ø. We also prove two approximation theorems for M.

Original languageEnglish
Pages (from-to)55-85
Number of pages31
JournalTransactions of the American Mathematical Society
Volume350
Issue number1
DOIs
StatePublished - 1998

Keywords

  • Absolute galois group
  • Field of totally 5-adic numbers
  • Global fields
  • Haar measure
  • Henselian fields
  • Local global principle
  • P5c fields over rings
  • Pac field over rings
  • Valuations

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