TY - JOUR

T1 - Rumely's local global principle for algebraic psc fields over rings

AU - Jarden, Moshe

AU - Razon, Aharon

N1 - Funding Information:
ACKNOWLEDGMENTS. Cryo-EM micrographs were collected at the David Van Andel Advanced Cryo-Electron Microscopy Suite at the Van Andel Research Institute. We thank Drs. Gongpu Zhao and Xing Meng for facilitating data collection. We are also grateful to Dr. Olga Yurieva (Rockefeller University) for WT Pol δ and Pol δexo- and to Dr. Nina Yao (Rockefeller University) for the Pol31-Pol32 complex. This work was supported by the NIH (GM131754, to H.L. and GM115809, to M.E.O.) and the Howard Hughes Medical Institute (M.E.O).

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

KW - Absolute galois group

KW - Field of totally 5-adic numbers

KW - Global fields

KW - Haar measure

KW - Henselian fields

KW - Local global principle

KW - P5c fields over rings

KW - Pac field over rings

KW - Valuations

UR - http://www.scopus.com/inward/record.url?scp=21944444792&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-98-01630-4

DO - 10.1090/s0002-9947-98-01630-4

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AN - SCOPUS:21944444792

VL - 350

SP - 55

EP - 85

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -